An Approach to Distributed Systems from Orderings and Representability

Abstract

In the present paper, we propose a new approach on ‘distributed systems’: the processes are represented through total orders and the communications are characterized by means of biorders. The resulting distributed systems capture situations met in various fields (such as computer science, economics and decision theory). We investigate questions associated to the numerical representability of order structures, relating concepts of economics and computing to each other. The concept of ‘quasi-finite partial orders’ is introduced as a finite family of chains with a communication between them. The representability of this kind of structure is studied, achieving a construction method for a finite (continuous) Richter–Peleg multi-utility representation.

1 Introduction and Motivation

In the present paper, we focus on an ordered structure known as distributed system. Although this concept belongs primarily to the field of computer science, its mathematical structure is common to many areas.

The representability issue appears too in a wide range of fields, such as economics and decision making [1, 2, 9, 12, 20, 28, 32, 34], computing [21, 26, 27, 29, 35, 38], and mathematical psychology [13,14,15, 31]. This interest on the representability of relations is usually due to maximization problems (in economics and decision making), the need to control non-linear processes that are being executed (computer science), the convenience of transform qualitative scales into quantitative ones (mathematical psychology), etc.

Hence, one of the goals of the present work is to bring to the computer field knowledge from other areas (order theory and decision theory in mathematics or economics, for instance) that may be helpful when dealing with distributed systems. For this purpose, first, we redefine the concepts of a distributed system. For that, we shall use the concept of biorder. This study was started in Ref. [16] for the particular case of distributed systems of two processes. However, in order to formalize completely the concept of distributed system, a further study is needed, linking biorders between n totally ordered sets.

The idea of a biorder was studied by Guttman (see Guttman Scales in Ref. [24]) and by Riguet under the name of Ferrers relation [37]. However, the concept was introduced for first time by André Ducamp and Jean-Claude Falmagne in 1969 in [15], and studied in depth in 1984 by Jean-Paul Doignon, André Ducamp and Jean-Claude Falmagne in Ref. [13]. It is defined as follows:

A biorder $\mathrel {{\mathcal {R}}}$ from A to X is a binary relation, with $\mathrel {{\mathcal {R}}}\subseteq A\times X$, satisfying that for every $a, b \in A$ and $x, y \in X$ $(a\mathrel {{\mathcal {R}}} x) \wedge (b\mathrel {{\mathcal {R}}} y)$ implies $(a\mathrel {{\mathcal {R}}} y) \vee (b\mathrel {{\mathcal {R}}} x)$.

The concept of biorder can also be found just as a Ferrers relation [2, 13], that is, as a relation $\mathrel {{\mathcal {R}}}$ on a single set X such that for any $x,y,z,t\in X$ it holds that $x\mathrel {{\mathcal {R}}} y$ with $z\mathrel {{\mathcal {R}}} t$ implies that $x\mathrel {{\mathcal {R}}} t$ or $z\mathrel {{\mathcal {R}}} y$. Other kind of relations such as interval orders [2, 5, 6, 9, 22, 23, 39, 41] (i.e., the more restrictive case of a semiorder [11, 17, 18, 28]), may be considered particular cases of biorders [2, 13].

In the present paper, we focus on biorders defined between disjoint totally ordered sets (see Fig. 1), and we use them in order to redefine the concept of distributed system [21, 27, 29, 35].

Fig. 1

A biordered pair of totally preodered sets

In a distributed system, distinct computers are connected to each other in order to achieve a common goal, this is known as distributed computing.

These computers communicate with each other through messages that are sent and received. Each computer has its own internal (physical) clock, so that it is possible to assign a number (a time) to each event of the process. Thus, from a mathematical point of view, each computer (or process) is a totally ordered set (i.e., a chain) which can be represented through its local time. Hence, without a precise clock synchronization, it is not possible to capture the causality relation between events of distinct processes. Moreover, if an event b holds later (with respect to the time) than a, it does not imply that a causally affects b. On the contrary, if a causally affects b, then b must hold later (with respect to an idyllic global time) than a. Finally, in these structures, the property called causal ordering of messages is usually¹ satisfied: if a computer i sends two messages $m_1$ and then $m_2$ (so, such that $m_1$ has been sent before $m_2$) to the same computer j ($i\ne j$), then message $m_1$ must be received before message $m_2$ (see Fig. 2). [26, 27]

Fig. 2

The vertical arrows represent the sequence of events of each process, i.e., the direction of the time. The dashed arrows represent the sending of messages, from the sender to the receiver. Here, the causal ordering of messages is not satisfied

The concept of a distributed system is usually defined as Lamport did [27]:

Definition 1.1

An event (illustrated by a point in Figs. 1 and 3) is a uniquely identified runtime instance of an atomic action of interest. It is an occurrence at a point in time, i.e., a happening at a cut of the timeline, which itself does not take any time. A process (illustrated by a vertical line in Figs. 1 and 3) is a sequence of totally ordered events, i.e., for any event a and b in a process, either a comes before b or b comes before a. A distributed system consists of a collection of distinct processes which are spatially separated, and which communicate with one another by exchanging messages (illustrated by red arrows in Fig. 1 and wavy arrows in Fig. 3). It is assumed that sending or receiving a message is an event.

Since each process consists of a sequence of events, each process is a totally ordered set, and the communication through messages between the processes will be defined by means of biorders.

Fig. 3

Illustration of a distributed system, taken from the paper [27] of Leslie Lamport

Moreover, this communication between processes defines a causal relation known as ‘causal precedence’ or ‘happened before’ relation [21, 27, 29, 35] (common in causality too [33], but now related to the theory of relativity, see also [10, 25]). This causal relation was defined by Leslie Lamport in [27] as follows (see also [26]):

Definition 1.2

The causal precedence (denoted by $\rightarrow$) on the set of events of a distributed system is the smallest relation satisfying the following three conditions:

• 1. If a and b are events in the same process, and a comes before b, then $a\rightarrow b$.

• 2. If a is the sending of a message by one process and b is the receipt of the same message by another process, then $a \rightarrow b$.

• 3. If $a \rightarrow b$ and $b \rightarrow c$ then $a \rightarrow c$.

If $a \rightarrow b$, then it is said that a causally precedes b.

This definition was introduced by Leslie Lamport in 1978 (see [27]) and it has been used until nowadays. In the present paper, we shall introduce a new definition of the concept through orderings. Graphically, $a\rightarrow b$ implies that there is a path of causality from event a to event b (moving in the direction of the arrows, see Figs. 1 or 3), i.e., b is reachable from a.

Two distinct events a and b are concurrent if $a \nrightarrow b$ and $b \nrightarrow a$, that is, if they in no way can causally affect each other, so in that case it is not known which event happened first. It is assumed that $\rightarrow$ is irreflexive ($a \nrightarrow a$ for any event a), so, in case there are no cycles, $\rightarrow$ is a strict partial ordering on the set of all events in the system.

Finally, we shall focus too on finite Richter–Peleg multi-utility representations. The construction of these (continuous) representations for a given preorder may be a hard problem. For this purpose, the study of the (continuous, in case the sets are endowed with topologies) representability of biorders defined between totally preordered sets seems a right approach in order to achieve a (continuous and finite) Richter–Peleg multi-utility representation of the corresponding causal precedence or happened before relation. This idea consists in using (continuous) representations of distributed systems in order to represent (continuously) the corresponding causal precedence relation.

The structure of the paper goes as follows:

After this introduction, a section of preliminaries is included. Next, in Sect. 3, a new definition (and its motivation) of a distributed system is introduced, so that then the representability problem is studied in Sect. 4, achieving an aggregation result for the case of line communications. In this section, it is shown how to construct weak representations of distributed systems with line communications starting from pairs of functions that represent each biorder. Finally, in Sect. 5, we focus on quasi-finite partial orders as an interesting family of partial orders. For this kind of orderings, we also include a technique in order to construct a finite (and continuous) Richter–Peleg multi-utility.

2 Notation and Preliminaries

From now on A, B and X as well as $X_1,\ldots , X_n$ will denote non-empty (maybe infinite) sets. When we speak of continuity of a real-valued mapping defined on a set S, we assume that some topology $\tau _S$ is given on S.

Definition 2.1

A binary relation $\mathrel {{\mathcal {R}}}$ from A to X is a subset of the Cartesian product $A \times X$. In particular, in the case that $A=X$, the binary relation $\mathrel {{\mathcal {R}}}$ is said to be defined on X, and it is a subset of the Cartesian product $X \times X$. Given two elements $a\in A$ and $x\in X$, we will use notation $a \mathrel {{\mathcal {R}}} x$ to express that the pair (x, y) belongs to ${\mathcal {R}}$. Associated to a binary relation $\mathrel {{\mathcal {R}}}$ from A to X, its negation is the binary relation $\mathrel {{\mathcal {R}}}^c$ from A to X given by $(a,x) \in \mathrel {{\mathcal {R}}}^c \iff (a,x) \notin \mathrel {{\mathcal {R}}}$ for every $a\in A$ and $x \in X$.

Given two binary relations $\mathrel {{\mathcal {R}}}$ and $\mathrel {{\mathcal {R}}'}$ on a set X, it is said that $\mathrel {{\mathcal {R}}'}$ extends or refines $\mathrel {{\mathcal {R}}}$ if $x \mathrel {{\mathcal {R}}} y$ implies $x \mathrel {{\mathcal {R}}'} y$, that is, if $\mathrel {{\mathcal {R}}}$ is contained in $\mathrel {{\mathcal {R}}'}.$

The transitive closure of a binary relation $\mathrel {{\mathcal {R}}}$ on a set X is the transitive relation $\mathrel {{\mathcal {R}}}^+$ on set X such that $\mathrel {{\mathcal {R}}}^+$ contains $\mathrel {{\mathcal {R}}}$ and $\mathrel {{\mathcal {R}}}^+$ is minimal.

The transitive reduction of a binary relation $\mathrel {{\mathcal {R}}}$ on a set X is, in case it exists, the smallest relation having the transitive closure of $\mathrel {{\mathcal {R}}}$ as its transitive closure.

Given a binary relation $\mathrel {{\mathcal {R}}}$ on X, if two elements $x,y\in X$ cannot be compared, that is, $\lnot (x \mathrel {{\mathcal {R}}} y)$ as well as $\lnot (y \mathrel {{\mathcal {R}}} x)$, then it is denoted by $x \bowtie y$. We shall denote $x{\mathcal {I}}y$ whenever $x{\mathcal {R}}y$ as well as $y{\mathcal {R}}x$.

Sometimes (depending on the ordering or when different relations are mixed) the standard notation is different. We also include it here.

Definition 2.2

A preorder $\precsim$ on X is a binary relation on X which is reflexive and transitive. An antisymmetric preorder is said to be an order. A total preorder $\precsim$ on a set X is a preorder such that if $x,y \in X$ then $[x \precsim y] \vee [y \precsim x]$. A total order is also called a linear order, and a totally ordered set $(X,\precsim )$ is also said to be a chain. Usually, an order that fails to be total is also said to be a partial order and it is also denoted by $\preceq$. A subset Y of a partially preordered set $(X, \precsim )$ is said to be an antichain if $x\bowtie y$ for any $x,y\in Y$.

If $\precsim$ is a preorder on X, then the associated asymmetric relation or strict preorder is denoted by $\prec$ and the associated equivalence relation by $\sim$ and these are defined, respectively, by $[x \prec y \iff (x \precsim y) \wedge \lnot (y \precsim x)]$ and $[x \sim y \iff (x\precsim y) \wedge (y \precsim x)]$. In the case of a finite partial order (also known as poset), it is quite common to denote $\precsim$ by $\sqsubseteq$ and $\prec$ by $\sqsubset$, respectively. The asymmetric part of a linear order (respectively, of a total preorder) is said to be a strict linear order (respectively, a strict total preorder).

Definition 2.3

A preorder $\precsim$ on X is said to be near-complete if $width(X, \precsim )$ $=n<\infty$. That is, if all antichains have cardinalities less or equal than n (for some $n\in {\mathbb {N}}$) as well as there is—at least—one antichain which cardinality is n.

Definition 2.4

A binary relation $\prec$ from A to X is a biorder if it is Ferrers, that is, if for every $a, b \in A$ and $x, y \in X$ the following condition holds:

$(a\prec x) \wedge (b\prec y) \Rightarrow (a\prec y) \vee (b\prec x).$

Related to $\prec$ we shall use the binary relation $\precsim$ from X to A given by $x\precsim a \iff \lnot (a\prec x)$, $a\in A, \, x\in X$. It is also common to use $\succsim$ from A to X given by $a\succsim x \iff \lnot (a\prec x)$, $a\in A, \, x\in X$.

Definition 2.5

Associated to a biorder $\prec$ defined from A to X, we shall consider two new binary relations [12, 13]. These binary relations are said to be the traces of $\prec$. They are defined on A and X, respectively, and denoted by $\prec ^*$, $\prec ^{**}$. They are defined as follows:

First, $a \prec ^* b \iff a \prec z \precsim b \$ for some $z \in X \ \ (a,b \in A),$ and, similarly, $x \prec ^{**} y \iff x \precsim c \prec y \$ for some $c \in A \ \ (x,y \in X).$

Remark 2.6

In the case of interval orders (so $A=X$), the binary relations denoted by $\prec ^*$ and $\prec ^{**}$ coincide with the “left trace” and “right trace” of the interval order. The names “left trace” and “right trace” have been used in the case of biorders too [8], and other notations such as $\prec _A$ and $\prec _X$ or $\prec ^l$ and $\prec ^r$ can be found in literature [8, 13, 31].

Remark 2.7

We set $a \precsim ^* b \iff \lnot (b \prec ^* a)$, $a \sim ^* b \iff a \precsim ^* b \precsim ^* a$, $x \precsim ^{**} y \iff \lnot (y \prec ^{**} x)$ and $x \sim ^{**} y \iff x \precsim ^{**} y \precsim ^{**} x$.

These weak relations can be characterized as follows [2, 5, 13]:

$$\begin{aligned}{} & {} a\precsim ^* b \iff \{b\prec x \Rightarrow a\prec x\}, \text { for any } x\in X. \\ {}{} & {} x\precsim ^{**} y \iff \{a\prec x \Rightarrow a\prec y\}, \text { for any } a\in A. \end{aligned}$$

As a matter of fact, both the binary relations $\precsim ^*$ and $\precsim ^{**}$ are total preorders on A and on X, respectively, if and only if the relation $\prec$ is a biorder [13]. Hence, in that case the indifference relations $\sim ^*$ and $\sim ^{**}$ are in fact equivalence relations so, it is possible to define the quotient set $A/\sim ^*$ and $X/\sim ^{**}$ [9, 13].

Next Definition 2.11 introduces the notion of representability² for total preorders and biorders. The goal of a representation is to convert a qualitative preference into a quantitative one, comparing real numbers instead of elements of a set.

Definition 2.8

Given a preorder $\precsim$ on X, a real function $u:X \rightarrow {\mathbb {R}}$ is said to be isotonic or increasing if for every $x,y \in X$ the implication $x\precsim y \Rightarrow u(x)\le u(y)$ holds true. In addition, if it also holds true that $x\prec y$ implies $u(x)< u(y)$, then u is said to be a Richter–Peleg utility representation.

A (not necessarily total) preorder $\precsim$ on a set X is said to have a multi-utility representation [20] if there exists a family ${\mathcal {U}}$ of isotonic real functions such that for all points $x,y \in X$ the equivalence $\{x \precsim y \Leftrightarrow \forall u \in {{\mathcal {U}}} \,\,(u(x) \le u(y))\}$ holds.

This kind of representation always exists for every preorder $\precsim$ on X (see Proposition 1 in [20]). It is also interesting to search for a continuous multi-utility representation of a preorder $\precsim$ when the set X is endowed with a topology $\tau$ (cf., for instance, [1, 20]), as well as for multi-utility representations through a finite number of functions.

When all the functions of the family ${\mathcal {U}}$ are order-preserving for the preorder $\precsim$ (i.e., for all $u \in {{\mathcal {U}}}$, and $x,y \in X$, $x \prec y$ implies that $u(x) < u(y)$), then the representation is called Richter–Peleg multi-utility representation [30].

In case of a poset, we shall use too the following concept³.

Definition 2.9

Let $(X,\sqsubseteq )$ be a finite partially ordered set with $|X|=n$. We shall say that a Richter–Peleg multi-utility representation ${\mathcal {U}}$ is bijective when each function $u\in {\mathcal {U}}$ is a bijection from X to $\{1,\ldots ,n\}$.

Remark 2.10

The number of functions needed for a Richter–Peleg multi-utility coincides with the dimension of the partial order [40].

Definition 2.11

A total preorder $\precsim$ on X is called representable if there is a real-valued function $u:X\rightarrow \mathbb R$ that is order-preserving, so that, for every $x, y \in X$, it holds that $[x \precsim y \iff u(x) \le u(y)]$. The map u is said to be an order-monomorphism (also known as a utility function for $\precsim$).

A biorder $\prec$ from A to X is said to be representable (as well as realizable with respect to <) if there exist two real-valued functions $u :A \rightarrow {\mathbb {R}}$, $v :X \rightarrow {\mathbb {R}}$ such that $a\prec x \iff u(a)<v(x)$ ($a\in A$, $x\in X$). In this case it is also said that the pair (u, v) represents $\prec$.

Although we will work with this definition of representability for biorders (realizable with respect to <), in Sect. 4 the following definition (introduced in [13]) is also needed:

A biorder $\prec$ from A to X is said to be representable with respect to $\le$ (as well as realizable with respect to $\le$) if there exist two real-valued functions $u :A \rightarrow {\mathbb {R}}$, $v :X \rightarrow {\mathbb {R}}$ such that $a\prec x \iff u(a)\le v(x)$ ($a\in A$, $x\in X$). In this case we shall say that the pair (u, v) represents $\prec$ with respect to $\le$.

Definition 2.12

Let $\prec$ be an asymmetric relation from a topological space $(A,\tau _A)$ to $(X,\tau _X)$. The relation $\prec$ is said to be $\tau _A$-continuous if the strict contour set $L_\prec (x) = \{ a \in A: a \prec x \}$ is a $\tau _A$-open set, for every $x \in X$. Dually, it is said to be $\tau _X$-continuous if the strict contour set $U_\prec (a) = \{ x \in X: a \prec x \}$ is a $\tau _X$-open set, for every $a \in A$. We shall say that the relation is bicontinuous if it is both $\tau _A$-continuous and $\tau _X$-continuous.

In particular, in the case of a single set X (that is, $A=X$) endowed with a single topology $\tau$ (so, $\tau =\tau _A=\tau _X$), the binary relation $\prec$ is said to be upper semi-continuous if the strict contour set $L_{\prec } (x) = \{ y \in X: y \prec x \}$ is $\tau$-open, for every $x \in X$. Dually, it is said to be lower semi-continuous if the strict contour set $U_{\prec } (x) = \{ y \in X: x \prec y \}$ is $\tau$-open, for every $x \in X$. We shall say that the relation is $\tau$-continuous if it is both upper and lower semi-continuous.

Definition 2.13

A biorder $\prec$ from A to X is said to be continuously representable on $(A,\tau _A)$ if it admits a representation (u, v) such that the function $u:A \rightarrow {\mathbb {R}}$ is continuous when A is given the topology $\tau _A$ and the real line is given its usual topology. Dually, $\prec$ is said to be continuously representable on $(X,\tau _X)$ if it admits a representation (u, v) such that the function $v:X \rightarrow {\mathbb {R}}$ is continuous when X is given the topology $\tau _X$ and the real line is given its usual topology. We say that the biorder is continuously representable if it admits a representation (u, v) such that both functions u and v are continuous [3].

An example of a biorder (also related to the traces and the continuity of the representation) may be found in [16]. Let us recall now some characterizations of the representability, for total preorders and biorders.

Definition 2.14

A total preorder $\precsim$ defined on X is said to be perfectly separable if there exists a countable subset $D \subseteq X$ such that for every $x,y \in X$ with $x \prec y$ there exists $d \in D$ such that $x \precsim d \precsim y$.

Let $\prec$ be a biorder from A to X. A subset M of $A\cup X$ is said to be strictly dense (see [2, 13]) if for all $a\in A$ and $x\in X$, $a\prec x$ implies the existence of an element $m\in M$ such that either $m\in X \text { and } a\prec m\precsim ^{**} x$, or $m\in A \text { and } a\precsim ^* m\prec x$.

Remark 2.15

Notice that if the number of classes defined by the equivalence relation $\sim ^*$ (or $\sim ^{**}$) of the trace $\precsim ^*$ (resp. $\precsim ^{**}$) on A (resp. X) is countable, then the biorder has a trivial strictly dense subset made by means of the representatives of classes $A/\sim ^*$ (resp. $X/\sim ^{**}$).

The following result is well known [9, 13].

Theorem 2.16

Let A and X be two non-empty sets.

• (a) A total preorder $\precsim$ on X is representable if and only if it is perfectly separable.

• (b) A biorder $\prec$ from A to X is representable if and only if there exists a countable strictly dense subset.

A similar study on representability of interval orders, semiorders (see also [11]) and total preorders but now in the extended real line $\bar{{\mathbb {R}}}$ appears in [17].

Given a biorder $\prec$ from A to X, it is possible to define the corresponding quotient sets as well as a new relation (see [2]) $\mathrel {{\widehat{\prec }}}$ from $A/\sim ^*$ to $X/\sim ^{**}$ by

$$\begin{aligned} {\widehat{a}}\mathrel {{\widehat{\prec }}} {\widehat{x}}\iff a\prec x, \text { for any } {\widehat{a}}\in A/\sim ^*, \, {\widehat{x}}\in X/\sim ^{**}. \end{aligned}$$

It is known (see [2]) that this relation $\mathrel {{\widehat{\prec }}}$ is well defined and that it is actually a biorder: the quotient biorder. It also holds true that ${\widehat{a}}\mathrel {{\widehat{\prec }}}^* {\widehat{b}}\iff a\prec ^* b$, and ${\widehat{x}}\mathrel {{\widehat{\prec }}}^{**} {\widehat{y}}\iff x\prec ^{**} y$, for any ${\widehat{a}}, {\widehat{b}}\in A/\sim ^*$ and for any ${\widehat{x}}, {\widehat{y}}\in X/\sim ^{**}$. Thus, any representation $({\widehat{u}},{\widehat{v}})$ of $\mathrel {{\widehat{\prec }}}$ has the additional property that ${\widehat{u}}$ and ${\widehat{v}}$ also represent the traces $\mathrel {{\widehat{\prec }}}^*$ and $\mathrel {{\widehat{\prec }}}^{**}$, respectively.

Moreover, any representation $({\widehat{u}},{\widehat{v}})$ of $\mathrel {{\widehat{\prec }}}$ delivers also a representation of $\prec$ just defining $u(a)={\widehat{u}}({\widehat{a}})$ and $v(x)={\widehat{v}}({\widehat{x}})$ (for any $a\in A$ and $x\in X$). Therefore, if $\mathrel {{\widehat{\prec }}}$ is representable then $\prec$ is representable, too. The converse is also true as the following lemma shows [2]:

Lemma 2.17

Let $\prec$ be a biorder from A to X and let $\mathrel {{\widehat{\prec }}}$ be the corresponding quotient biorder from $A/\sim ^*$ to $X/\sim ^{**}$. Assume that $\prec$ admits a representation (u, v). Then, $\mathrel {{\widehat{\prec }}}$ is also representable.

The following definition was introduced by Nakamura in [31] and, as it is shown in Corollary 2.19 (see also [31] or Remark 1 in [8]), it is equivalent to the aforementioned order-denseness condition named ‘strictly dense’.

Definition 2.18

Let $\prec$ be a biorder from A to X. A pair of subsets $A^*\subseteq A$ and $X^*\subseteq X$ is said to be jointly dense for $\prec$ if for all $a\in A$ and $x\in X$, $a\prec x$ implies the existence of two elements $a^*\in A^*$ and $x^*\in X^*$ such that $a\precsim ^* a^*\prec x^*\precsim ^{**} x$.

Next corollary is well known [2, 8, 12, 31]:

Corollary 2.19

Let $\prec$ be a biorder from A to X. The following statements are equivalent:

• (i) The biorder has a pair of jointly dense and countable subsets.

• (ii) The biorder has a countable strictly dense subset.

• (iii) The biorder is representable.

• (iv) The biorder is representable through a pair of functions (u, v) with the additional condition that u represents the trace $\precsim ^*$ and v the trace $\precsim ^{**}$.

3 A New Definition for Distributed System

In the following pages, we introduce a new definition for the concept of a distributed system with n processes (the original definitions of these concepts—an event, a process and a distributed system—have been introduced in Sect. 1). But first, we redefine the concepts of causal precedence and communication.

Since in the following pages many relations are going to appear, for the sake of clarity, from now, we shall use the symbol ${\mathcal {P}}$ in order to refer to a biorder relation, whereas we shall use the symbol $\precsim$ for total preorders (also for the traces associated to a biorder).

Definition 3.1

Let $\{(X_k, \prec _k)\}_{k\in K}$ be a finite family of strict partially ordered and disjoint sets (i.e., $X_i\cap X_j=\emptyset , \, \forall \, i\ne j$) and $\{{\mathcal {P}}_{ij}\}_{i\ne j}$ a family of relations from $X_i$ to $X_j$ (for any $i\ne j, \, i,j\in K$). The causal precedence corresponding to $\{\prec _k\}_{k\in K}\cup \{{\mathcal {P}}_{ij}\}_{i\ne j}$ on $X=\bigcup \limits _{k\in K} X_k$ is the transitive closure of the union $(\bigcup \limits _{k\in K} \prec _k) \bigcup (\bigcup \limits _{i\ne j}\{{\mathcal {P}}_{ij}\})$. This relation shall be denoted by $\rightarrow$:

$$\begin{aligned} \left\{ \left( \bigcup \limits _{k\in K} \prec _k\right) \bigcup \left( \bigcup \limits _{i\ne j}\left\{ {\mathcal {P}}_{ij}\right\} \right) \right\} ^+= \rightarrow . \end{aligned}$$

Remark 3.2

Notice that, with the definition above, the absence of cycles is not guaranteed, as it is shown the Fig. 4. The existence of this cycles implies an error in the computing, known as deadlock [26].

Fig. 4

A cycle in a distributed system: a deadlock

If there is no cycle, then the transitive closure is a strict partial order. In the present paper, we will assume that there is no error or deadlock in the distributed system, that is, we shall assume that the causal precedence is a strict partial order.

Now, recovering the idea of a distributed system in the spirit of Definition 1.1, we first mathematically formalize the idea of communication just as a finite relation between to distinct sets satisfying a ‘bijective’ condition.

Definition 3.3

Let A and B be two disjoint sets. A communication from A to B is a finite binary relation ${\mathcal {P}}\subseteq A\times B$ (i.e., $|{\mathcal {P}}|<\infty$) such that for any $(a,b)\in {\mathcal {P}}$ and $a'\in A, \, b'\in B$ the following bijective condition is satisfied:

$$\begin{aligned} (a',b)\in {\mathcal {P}}\Rightarrow a=a' \quad \text { as well as }\quad (a,b')\in {\mathcal {P}}\Rightarrow b'=b. \end{aligned}$$

Here, the elements $a\in A$ such that $(a,b)\in {\mathcal {P}}$ (for some $b\in B$) are said to be the senders, whereas the elements $b\in B$ such that $(a,b)\in {\mathcal {P}}$ (for some $a\in A$) are said to be the receivers.⁴

Now, we focus on communications between ordered sets.

Definition 3.4

Let $(A,\precsim _A)$ and $(B,\precsim _B)$ be two disjoint partially ordered sets. We shall say that a binary relation ${\mathcal {P}}$ is a causal biorder from A to B if ${\mathcal {P}} \subseteq A\times B$ and for any $a,c\in A$, $b,d\in B$ it holds that

$$\begin{aligned} (a{\mathcal {P}} b) \text { and } (c{\mathcal {P}} d)\quad \Rightarrow \quad (a\rightarrow d) \text { or } (c\rightarrow b), \end{aligned}$$

where $\rightarrow$ denotes the corresponding causal precedence of $\precsim _A \cup \precsim _B \cup \mathrel { {\mathcal {P}}}$ on $A\cup B$.

Example 3.5

Let $(A,\precsim _A)$ and $(B,\precsim _B)$ be two partially ordered sets as defined in Fig. 5. Let ${\mathcal {P}}$ and ${\mathcal {P}}'$ be two relations from A to B defined by

$$\begin{aligned} {\mathcal {P}}=\{(a_2, b_1), (a_3,b_3)\} \quad \text { and } \quad {\mathcal {P}}'=\{ (a_2,b_2), (a_3,b_3)\}. \end{aligned}$$

It is straightforward to see that ${\mathcal {P}}$ is a causal biorder whereas ${\mathcal {P}}'$ it is not.

Fig. 5

Two partially ordered sets, $(A,\precsim _A)$ and $(B,\precsim _B)$

Definition 3.6

Let $(A,\precsim _A)$ and $(B,\precsim _B)$ be two disjoint partially ordered sets and ${\mathcal {P}}$ a communication from A to B. We define the relation $\overline{{\mathcal {P}}}$ from A to B by

$$\begin{aligned} a\overline{{\mathcal {P}}}b \iff a\precsim _A a'{\mathcal {P}} b' \precsim _B b, \text { for some } a'\in A, b'\in B. \end{aligned}$$

That is, $\overline{{\mathcal {P}}}=\precsim _{A}\circ {\mathcal {P}}\circ \precsim _{B}$.

Proposition 3.7

Let $(A,\precsim _A)$ and $(B,\precsim _B)$ be two disjoint partially ordered sets and ${\mathcal {P}}$ any relation from A to B. If $(A, \precsim _A)$ or $(B, \precsim _B)$ is a chain, then:

• (i) ${\mathcal {P}}$ is a causal biorder.

• (ii) $\overline{{\mathcal {P}}}$ is a biorder.

In particular, any communication from A to B is a causal biorder.

Proof

Let $a,x\in A$ and $b,y\in B$ be elements such that $a{\mathcal {P}} b$ and $x{\mathcal {P}} y$. If $(A, \precsim _A)$ is a chain, then $a\precsim _A x$ or $x\precsim _A a$ is satisfied. Hence, it holds that $a\precsim _A x {\mathcal {P}} y b$ or $x\precsim _A a {\mathcal {P}} b$, thus, $a \rightarrow b$ or $x\rightarrow b$. Therefore, ${\mathcal {P}}$ is a causal biorder. We reason dually if $(B,\precsim _B)$ is a chain.

Let $a,b\in A$ and $x,y\in B$ be points such that $a \overline{{\mathcal {P}}} x$ and $b \overline{{\mathcal {P}}} y$. Hence, by definition, there exist $a',b'\in A$ and $x',y'\in X$ such that $a\precsim _A a' {\mathcal {P}} x' \precsim _B x$ and $b\precsim _A b' {\mathcal {P}} y' \precsim _B y$. If $(A, \precsim _{A})$ is a chain, we distinguish two cases:

• 1. If $a\precsim _A b$, then $a\precsim _A b' {\mathcal {P}} y' \precsim _B y$, so $a\overline{{\mathcal {P}}} y$.

• 2. If $b\precsim _A a$, then $b\precsim _A a' {\mathcal {P}} x' \precsim _B x$, so $b\overline{{\mathcal {P}}} x$.

Therefore, $\overline{{\mathcal {P}}}$ is a biorder. We reason dually in case $(B, \precsim _{B})$ is a chain. $\square$

Remark 3.8

• (1) By Proposition 3.7, it is clear that a communication ${\mathcal {P}}$ is a causal biorder as well as $\overline{{\mathcal {P}}}$ is a biorder.

• (2) In order to keep close to the original definition given by Lamport (see Definition 1.1), where sending or receiving a message is an event, in Definition 3.3 is not allowed to send a message from a in A to more than one receiver in B. Dually, an element b in B cannot receive more than one message from A. From a mathematical point of view, it would be possible to generalize the idea of communication without restricting it to a finite cardinal, i.e., with $|{\mathcal {P}}|=\infty$. However, in the present paper, we shall work just on communications in the sense of Definition 3.3.

Now we are ready to propose a definition of a distributed system of n sets from a mathematical and theoretical point of view (the original definitions of these concepts—an event, a process and a distributed system—have been introduced in Sect. 1).

Definition 3.9

Let $\{(X_i,\precsim _i)\}_{i=1}^n$ be a family of disjoint and totally ordered sets and $\mathrel {{\mathcal {P}}}=\{\mathrel {{\mathcal {P}}_{ij}}\}_{i\ne j}$ (with $i,j\in \{1,\ldots ,n\}$) be a family of communications from $X_i$ to $X_j$ (with $i\ne j$). Each totally ordered set is said to be a process. Each element of the processes is said to be an event. The pair $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}})$ is said to be a distributed system.

Remark 3.10

• (1) In the previous definition, as it was in the original definition of L. Lamport (see Ref. [27], in particular page 559 and footnote 2), the messages may be received out of order (i.e., without satisfying the causal ordering of messages). Furthermore, with this definition there may be cycles with respect to the causal relation (see Fig. 4).

• (2) Notice that a communication may be empty, so that there is no sending of messages in one direction between two processes. In this case, we shall denote $\mathrel {{\mathcal {P}}_{ij}}=\{\emptyset \}$.

• (3) It can be proved that each total order $\precsim _i$ in $X_i$ refines the traces $\precsim _{ij}^*$ and $\precsim _{ji}^{**}$ related to the biorders $\overline{{\mathcal {P}}}_{ij}$ and $\overline{{\mathcal {P}}}_{ji}$ (respectively), for any $j\ne i$. Actually, this property was used in Ref. [16] in order to define the concept of a distributed system of two processes. However, the use of the communication concept in order to define a distributed system is closer to reality, since it captures the idea of sending and receiving messages. Moreover, it derives in the new term of causal biorder, which seems interesting when dealing with a set endowed with more than one relation. Hence, Definition 3.9 has been written by means of communications.

Proposition 3.11

Let $(A,\precsim _A)$ and $(B,\precsim _B)$ be two disjoint totally ordered sets and ${\mathcal {P}}_1$ and ${\mathcal {P}}_2$ two communications from A to B. Let $((A,\precsim _A)\cup (B,\precsim _B), {\mathcal {P}}_1)$ and $((A,\precsim _A)\cup$ $(B,\precsim _B), {\mathcal {P}}_2)$ be the corresponding distributed systems and $\rightarrow _1$ and $\rightarrow _2$ their causal relations, respectively. If the causal relations concur, i.e., $\rightarrow _1=\rightarrow _2$, then the communications are also the same (i.e., ${\mathcal {P}}_1={\mathcal {P}}_2$) or the causal ordering of messages is not satisfied.

Proof

Let $a\in A$ and $b\in B$ be such that $a{\mathcal {P}}_1 b$. Then, $a\rightarrow _1 b$, that means $a\rightarrow _2 b$, i.e., there exist $a_1,b_1$ such that $a\precsim _A a_1{\mathcal {P}}_2 b_1\precsim _B b$. Thus, $a{\mathcal {P}}_1 b$ implies $a\overline{{\mathcal {P}}}_2 b$.

Suppose now that $a\overline{{\mathcal {P}}}_2 b$ but $\lnot ( a {\mathcal {P}}_2 b)$. Then, there must exist $a_2\in A, \,b_2\in B$ such that $a\prec _A a_2{\mathcal {P}}_2 b_2\precsim _B b$ or $a\precsim _A a_2{\mathcal {P}}_2 b_2\prec _B b$. Assume that $a\prec _A a_2{\mathcal {P}}_2 b_2\precsim _B b$ is satisfied (we reason analogously for the dual case), then it holds that $a_2\rightarrow _2 b$ with $a\prec _A a_2$, that is, $a_2\rightarrow _1 b$ with $a\prec _A a_2$. Therefore, $a_2\overline{{\mathcal {P}}}_1 b$, i.e., $a\prec a_2\precsim _A a_3{\mathcal {P}}_1 b_3\precsim _B b$ for some $a_3\in A$, $b_3\in B$.

Here, we distinguish two cases. If $b_3=b$, then ${\mathcal {P}}_1$ fails to be a communication since we have that $a_3{\mathcal {P}}_1 b$ as well as $a{\mathcal {P}}_1 b$, with $a\ne a_3$. If $b_3\prec b$, then the causal ordering of messages is not satisfied, since we have that $a_3{\mathcal {P}}_1 b_3$ and $a {\mathcal {P}}_1 b$ as well as $a\prec _A a_2$ and $b_3\prec _B b$ (see Fig. 2). This concludes the proof. $\square$

Now, we introduce the concept of line communication.

Definition 3.12

Let $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}})$ be a distributed system, where $\mathrel {{\mathcal {P}}}=\{\mathrel {{\mathcal {P}}_{ij}}\}_{i\ne j}$ (with $i,j\in \{1,\ldots ,n\}$) is the family of communications from $X_i$ to $X_j$ (with $i\ne j$). It is said that $\mathrel {{\mathcal {P}}}$ is a line communication if ${\mathcal {P}}_{ij}=\{\emptyset \}$ for any $j\ne i+1$, for each $i=1,\ldots ,n-1$.

Hence, when the processes are endowed with a line communication, these computers or processes are ordered in a sequence (i.e., totally ordered) such that each computer only sends messages to the next one (see Fig. 6).

Fig. 6

A distributed system of three processes with line communication

Proposition 3.13

Let $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$ and $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}'}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}'_{ij}} )$ be two distributed systems with the same processes and both with line communication. Assume that the causal ordering of messages is always satisfied as well as there is no cycles (i.e., the $\rightarrow$ is a strict partial order). Then, the corresponding causal precedences coincide ($\rightarrow _1=\rightarrow _2$) if and only if $\mathrel {{\mathcal {P}}}_{ij}=\mathrel {{\mathcal {P}}'}_{ij}$ for each $i\ne j$, that is, if and only if they have the same communications.

Proof

$\Rightarrow :$ If the corresponding causal precedences coincide, then they also coincide when we restrict the relation to a subset $X_i\cup X_{i+1}$ (for any $i=1,\ldots ,n-1$). Moreover, since $\mathrel {{\mathcal {P}}}$ is a line communication, it holds that $\rightarrow _{1{|_{X_i\cup X_{i+1}}}}=(\precsim _i \cup \precsim _{i+1}\cup \mathrel {{\mathcal {P}}}_{i\, i+1})^+$, that is, the restriction of the causal relation $\rightarrow _1$ to $X_i\cup X_{i\, i+1}$ is just the causal relation of the distributed system $((X_i,\precsim _i)\cup (X_{i+1},\precsim _{i+1}), {\mathcal {P}}_{i\, i+1})$. Dually, it holds that $\rightarrow _{2{|_{X_i\cup X_{i+1}}}}$$=(\precsim _i \cup \precsim _{i+1}\cup \mathrel {{\mathcal {P}}}_{i\, i+1}')^+$. Therefore, by Proposition 3.11, the communications $\mathrel {{\mathcal {P}}}_{i\,i+1}$ and $\mathrel {{\mathcal {P}}'}_{i\, i+1}$ also coincide, and that for any $i=1,\ldots ,n-1$.

$\Leftarrow :$ This implication is trivial. $\square$

Remark 3.14

Notice that the decomposition of a partially ordered set in n disjoint chains is not unique. Therefore, it may be possible to construct distinct distributed systems such that the corresponding causal relation coincides with the initial partial order. In order to show that we include the following example:

Example 3.15

Let $\precsim$ be a partial order defined on $X=\{a,b,c,d\}$ by $\{ c\precsim b\precsim a, b\precsim d \}$. Then, the partial order can be characterized by means of the following distributed systems (among others) of 2 processes (see Fig. 7):

• (1) $(X_1,\precsim _1)= (\{ a,b,c\}, c\prec _1 b\prec _1 a)$ and $(X_2, \precsim _2) =(\{d\}, \{\emptyset \} )$, with communication ${\mathcal {P}}_{12}=\{(b, d)\}$.

• (2) $(X_1,\precsim _1)= (\{ a,c\}, c\prec _1 a)$ and $(X_2, \precsim _2) =(\{b,d\}, b\prec _2 d)$, with communications ${\mathcal {P}}_{12}=\{(c, b)\}$ and ${\mathcal {P}}_{21}=\{(b, a)\}$.

Fig. 7

A partial order represented through two distinct distributed systems

Remark 3.16

Hence, given a preorder, it seems interesting to study the existence and uniqueness of distributed systems that characterize (through its causal precedence) the preorder with some additional properties such as that the length of the processes is minimal, or that the number of messages is minimal. For instance, in the example of Fig. 7, in the first case, the length of the longest process is three and the number of messages is one, whereas in the second case, these values are two and two, respectively.

4 Representability of Distributed Systems

Since it is possible to add a new process to a distributed system (that is, connecting another computer to the system, including also the corresponding communication), it is interesting to study how to create a new representation of the distributed system that arise from the union of two distributed systems, but now aggregating the representations before.

In this paper we do not achieve the answer to this question but, at least, we are able to construct weak representations of distributed systems with line communications starting from pairs of functions that represent each biorder.

We shall assume that the causal ordering of messages is satisfied, as well as there are no cycles. Thus, we assume that our distributed systems are such that the causal relation $\rightarrow$ is a strict partial order.

The following definitions introduce the concept of representability for a distributed system of n processes.

Definition 4.1

Let $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$ be a distributed system. We say that it is weakly representable if there exists a family $\{u_i\}_{i=1}^n$ (called weak representation) of real functions $u_i:X_i\rightarrow {\mathbb {R}}$ such that $(u_i,u_j)$ weakly represents the biorder $\mathrel {\overline{{\mathcal {P}}}_{ij}}$ with respect to < (that is, $x_i \mathrel {\overline{{\mathcal {P}}}_{ij}} x_j \Rightarrow u_{i}(x_i)<u_{j}(x_j)$, for any $x_i\in X_i, \, x_j\in X_j$) as well as each $u_i$ represents the total order $\precsim _i$ (i.e., $x\precsim _i y \iff u_i(x)\le u_i(y), \, x,y\in X_i$), for any $i,j\in \{1,\ldots ,n\}$ and $i\ne j$.

If each set $X_i$ is endowed with a topology $\tau _i$, then we will say that the distributed system is continuously weakly representable if there exists a continuous weak representation.

Remark 4.2

• (1) Notice that, for any $x\in X_i$ and $y\in X_j$ such that $x\rightarrow y$, it holds that $u_i(x)<u_j(y)$, for any $i,j\in \{1,2,\ldots ,n\}$.

• (2) The aforementioned functions $u_i$ are known as Lamport clocks (see [27]). In fact, a Lamport clock is a function ${\mathcal {C}}$ satisfying that $a\rightarrow b$ implies ${\mathcal {C}}(a)<{\mathcal {C}}(b)$, for any $a,b\in X$. On the other hand, ${\mathcal {C}}(a)<{\mathcal {C}}(b)$ does not imply $a\rightarrow b$.

Definition 4.3

Let $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$ be a distributed system. We say that it is (finitely) representable if there exists a (finite) family of weak representations $\{ \{u_{i}^k\}_{i=1}^n\}_{k\in {\mathcal {K}}}$ such that $x_i \mathrel {\overline{{\mathcal {P}}}_{ij}} x_j$ iff $u^k_{i}(x_i)<u_{j}^k(x_j)$ for any $k\in {\mathcal {K}}$, for any $x_i\in X_i, \, x_j\in X_j$ and $i\ne j$.

If each set $X_i$ is endowed with a topology $\tau _i$, then we will say that the distributed system is continuously representable if there exists a continuous representation.

As following Preposition 4.4 shows, the term before is analogous to the Richter–Peleg multi-utility representation used for preorders. [4, 7, 20, 34, 36]

Proposition 4.4

A distributed system $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$ is (finitely) representable if and only if the corresponding causal relation $\rightarrow$ is (finitely) Richter–Peleg multi-utility representable.

Proof

Given a representation $\{ \{u_{i}^k\}_{i=1}^n\}_{k\in {\mathcal {K}}}$ of the distributed system, the family of functions $\{w_k\}_{k\in {\mathcal {K}}}$ defined by

$$\begin{aligned} w_k(x)=u_{i}^k(x) \text { if } x\in X_i, \text { with } k\in {\mathcal {K}},\end{aligned}$$

is a Richter–Peleg multi-utility representation of the strict partial order $\rightarrow$.

Dually, starting from a Richter–Peleg multi-utility representation $\{ w_k(x)\}_{k\in {\mathcal {K}}}$ of the causal relation $\rightarrow$ of a distributed system $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$, then the following representation $\{ \{u_{i}^k\}_{i=1}^n\}_{k\in {\mathcal {K}}}$ arises:

$$\begin{aligned} u_{i}^k(x)=w_k(x), \text { when } x\in X_i, \text { with } k\in {\mathcal {K}}, \text { for each } i=1,\ldots ,n. \end{aligned}$$

$\square$

Remark 4.5

Notice that the main difference is the domain of the corresponding functions. In the case of distributed systems, the functions are defined on the processes, whereas in the case of preorders they are defined on the all set (which would be the union of the processes). This difference may be relevant when dealing with continuity and topological spaces.

Now we introduce another kind of representation in bijection with the concept of multi-utility [4, 7, 20, 34, 36], and that it is actually common and known in computing by vector clock representation (see [29, 35]). Due to that coincidence (and in order to distinguish it from the definition before), we shall call it by vector representation.

Definition 4.6

Let $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$ be a distributed system. We say that it is vector representable if there exists a family of weak representations with respect to $\le$ (called vector clocks) $\{ \{u_{i}^k\}_{i=1}^n\}_{k\in {\mathcal {K}}}$ such that $x_i \mathrel {\overline{{\mathcal {P}}}_{ij}} x_j$ iff $u_{i}^k(x_i)\le u_{j}^k(x_j)$ for any $k\in {\mathcal {K}}$, as well as there exists an index $l\in {\mathcal {K}}$ such that $u_{i}^l(x_i)< u_{j}^l(x_j)$ (for any $x_i\in X_i, \, x_j\in X_j$ and $i\ne j$).

If each set $X_i$ is endowed with a topology $\tau _i$, then we will say that the distributed system is continuously vector representable if there exists a continuous vector representation.

Remark 4.7

In computing, these vector clocks are constructed through timestamps algorithms (see [29, 35]).

The problem of aggregating representations is not trivial. In order to illustrate that, we include the following example.

Example 4.8

Let ${\mathcal {P}}_{12}$ and ${\mathcal {P}}_{23}$ two communications between $A=\{a,b\}$ (with $a\prec _A b$) and $X=\{x,y\}$ (with $x\prec _X y$) and $\Lambda =\{\alpha , \beta \}$ (with $\alpha \prec _{\Lambda } \beta$), respectively, defined as follows:

$$\begin{aligned} a{\mathcal {P}}_{12} x{\mathcal {P}}_{23} \beta , \quad a{\mathcal {P}}_{12} y \quad \text{ and }\quad b{\mathcal {P}}_{12} y. \end{aligned}$$

Now we define the tuple (u, v, w) by $u(a)=0, u(b)=1, v(x)=1, v(y)=2, w(\alpha )=1$ and $w(\beta )=2$. Then, the pairs (u, v) and (v, w) are representations of the distributed systems defined on $A\cup X$ and on $X\cup \Lambda$, respectively. However, the tuple (u, v, w) fails to represent the distributed system made up by the three processes: $u(a)=0<w(\alpha )=1$ but $\lnot (a\rightarrow \alpha )$ (Fig. 8).

Fig. 8

A distributed system. Each process contains just two events. The dashed arrows represent the communication between processes

In the following lines, it is shown how to construct weak representations of distributed systems with line communications starting from pairs of functions that represent each biorder. For more clearness, before introduce the general case, first we include here the particular case of a distributed system of three processes with a linear communication.

Proposition 4.9

Let $(\bigcup _{i=1}^3 (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i=1 }^{2} \mathrel {{\mathcal {P}}_{i\, i+1}} )$ a distributed system of three processes with a linear communication such that for every $i\in \{1,2\}$ the pair $(u_i,v_i)$ is a representation of the biorder $\mathrel {\overline{{\mathcal {P}}}_{i\,i+1}}$.⁵ Then, the tuple $(u=u_1+u_{u_2}, v=v_1+u_2,w=v_2+v_{v_1})$ is a weak representation of the distributed system, where $u_{u_2}$ and $v_{v_1}$ are defined as follows:

$$\begin{aligned}{} & {} u_{u_2}(x)=\inf \{u_2(y):x\overline{{\mathcal {P}}}_{12} y\,; \, y\in X_2\}; \text { for any } x\in X_1,\\{} & {} v_{v_1}(x)=\sup \{v_1(y):y\overline{{\mathcal {P}}}_{23} x\,; \, y\in X_2\}; \text { for any } x\in X_3, \end{aligned}$$

on the assumption that $\inf \{\emptyset \}=1$ and $\sup \{\emptyset \}=0.$

Proof

First, notice that since the utilities take values on (0, 1), and taking into account that $\inf \{\emptyset \}=1$ and $\sup \{\emptyset \}=0$, the functions $u_{u_2}$ and $v_{v_1}$ are well defined (that is, the infimum and the supremum always exist).

Let x, y be two elements such that $x\overline{{\mathcal {P}}}_{12} y$. Then, it holds true that $u_1(x)<v_1(y)$ as well as $u_{u_2}(x)\le u_2(y)$. Therefore, the condition $u(x)<v(y)$ is satisfied. We argue analogously for a pair of elements x, y such that $x\overline{{\mathcal {P}}}_{23} y$.

Since $x\precsim _1 x' \overline{{\mathcal {P}}}_{12} y$ implies $x\overline{{\mathcal {P}}}_{12} y$, the inequality $u_{u_2}(x)\le u_{u_2}(x')$ is also satisfied. If $x\precsim _1 x'$ and there is no $y\in X_2$ such that $x' \overline{{\mathcal {P}}}_{12} y$, then $u_{u_2}(x')=\inf \{\emptyset \}=1$ so, again, it holds that $u_{u_2}(x)\le u_{u_2}(x')$. Thus, we conclude that $u_{u_2}(x)\le u_{u_2}(x')$ is always satisfied for any $x,x'\in X_1$ such that $x\precsim _1 x'$. We argue analogously for a pair of elements $y,y'\in X_3$ such that $y\precsim _{3} y'$. In addition, the functions $v_1$ and $u_2$ also represent the total order $\precsim _2$, as well as $u_1$ and $v_2$ represent the total orders $\precsim _1$ and $\precsim _3$, respectively. Hence, we deduce that the functions $u=u_1+u_{u_2}, v=v_1+u_2$ and $w=v_2+v_{v_1}$ are representations of the total orders $\precsim _1$, $\precsim _2$ and $\precsim _3$, respectively.

Let x, z be now two elements such that $x\overline{{\mathcal {P}}}_{12} y \overline{{\mathcal {P}}}_{23} z$, for some $y\in X_2$. Then, since it holds true that $u_1(x)<v_1(y)\le v_{v_1}(z)$ and $u_{u_2}(x)\le u_2(y)\le v_{2}(z)$, the condition $u(x)<w(z)$ is satisfied.

Thus, we conclude that $(u=u_1+u_{u_2}, v=v_1+u_2,w=v_2+v_{v_1})$ is a weak representation of the distributed system. $\square$

Before generalize the proposition above to n processes, first we introduce the following operators.

Definition 4.10

Let $(\bigcup _{i=1}^2 (X_i, \precsim _i), \mathrel {{\mathcal {P}}} )$ be a distributed system with a single communication ${\mathcal {P}}$ from $X_1$ to $X_2$. Let u and v be two (not necessarily strictly) increasing functions on $X_1$ and $X_2$ (respectively) that take values on (0, 1). Assume that $\inf \{\emptyset \}=1$ and $\sup \{\emptyset \}=0.$ Then, we define the following two operators:

$$\begin{aligned}{} & {} {\underline{op}}(v)(x)=\inf \{v(y) :x\overline{{\mathcal {P}}} y\}_{\{y\in X_2\}}\,; \quad x\in X_1,\\{} & {} {\overline{op}}(u)(x)=\sup \{u(y) :y\overline{{\mathcal {P}}} x\}_{\{y\in X_1\}}\,; \quad x\in X_2. \end{aligned}$$

We shall call these operators lower operator and upper operator, respectively.

Remark 4.11

• (1) Notice that, starting from a function u on $X_1$, ${\overline{op}}(u)$ defines a new function on $X_2$, and not in $X_1$. Dually, starting now from a function v on $X_2$, ${\underline{op}}(v)$ defines a new function on $X_1$, and not in $X_2$.

• (2) In fact, since sending and receiving messages is an event, the infimum (of a non-empty set) is a minimum and the suprema (of a non-empty set) is a maximum. Otherwise, the infimum of an empty set is the top of the ordered set, that is, 1, and the supremum is the bottom, i.e., 0.

Proposition 4.12

Let $(\bigcup _{i=1}^2 (X_i, \precsim _i), \mathrel {{\mathcal {P}}} )$ be a distributed system with a single communication ${\mathcal {P}}$ from $X_1$ to $X_2$. Let u and v be two (not necessarily strictly) increasing functions on $X_1$ and $X_2$ (respectively) that take values on (0, 1). Then, the following properties are satisfied:

• (i) ${\underline{op}}(v)$ and ${\overline{op}}(u)$ are increasing in $X_1$ and $X_2$, respectively.

• (ii) The pairs $({\underline{op}}(v), v )$ and $(u, {\overline{op}}(u))$ represent the biorder $\overline{{\mathcal {P}}}$ with respect to $\le$.

• (iii) $x_1\sim ^*y_1$ implies ${\underline{op}}(v)(x_1)={\underline{op}}(v)(y_1)$, as well as $x_2\sim ^{**}y_2$ implies ${\overline{op}}(v)(x_2)={\overline{op}}(v)(y_2)$, for any $x_1,y_1\in X_1, \, x_2,y_2\in X_2$.⁶

Proof

• (i) If $x\precsim _1 y$, then it holds that $y\overline{{\mathcal {P}}} z$ implies $x\overline{{\mathcal {P}}} z$. Therefore, applying the definition of the lower operator, it follows that the inequality ${\underline{op}}(v)(x)\le {\underline{op}}(v)(y)$ is satisfied, that is, ${\underline{op}}(v)$ is increasing with respect to $\precsim _1$ on $X_1$. We argue dually in order to prove that ${\overline{op}}(u)$ is increasing with respect to the total order $\precsim _{2}$ defined on $X_{2}$.

• (ii) If $x\overline{{\mathcal {P}}} y$, then applying the definition of the lower operator, it is clear that the inequality ${\underline{op}}(v)(x)\le (v)(y)$ is satisfied. We argue analogously for the pair $(u, {\overline{op}}(u))$. On the other hand, suppose that ${\underline{op}}(v)(x)\le v(y)$. Since v takes values on (0, 1), ${\underline{op}}(v)(x)=r\in (0,1)$, which means that (by definition of ${\underline{op}}(v)$) there exists $z\in X_2$ such that $x\overline{{\mathcal {P}}} z$ with $v(z)=r$ (here take into account Remark 4.11 (2)). Thus, $v(z)=r\le v(y)$ and then, $z\precsim _2 y$. Therefore, we conclude that $x\overline{{\mathcal {P}}} y$. Hence, the pair $({\underline{op}}(v), v )$ represents the biorder with respect to $\le$.

• (iii) If $x_1\sim ^*y_1$, then $x_1\overline{{\mathcal {P}}} z$ holds if and only if $y_1\overline{{\mathcal {P}}} z$ is satisfied, for any $z\in X_2$. Therefore, by the definition of the lower operator, the equality ${\underline{op}}(v)(x_1)= {\underline{op}}(v)(y_1)$ holds true. We argue analogously for the indifference $\sim ^{**}$.

$\square$

Remark 4.13

• (1) Dealing with a distributed system $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j } \mathrel {{\mathcal {P}}_{ij}} )$ of n processes, since—by Proposition 4.12 (i)—${\underline{op}}(u_i)$ and ${\overline{op}}(u_i)$ are increasing in their corresponding sets ($X_{i-1}$ and $X_{i+1}$, respectively), it is possible to apply an operator more than once. Therefore, starting from an increasing function $u_i$ in $X_i$, we shall denote by ${\underline{op}}^2(u_i)$ the function ${\underline{op}}({\underline{op}}(u_i))$ defined in $X_{i-2}$. This notation is generalized to ${\underline{op}}^k(u_i)$, achieving a function in $X_{i-k}$. We shall use the same notation for the upper operator ${\overline{op}}$. Since the hypothesis of Proposition 4.12 are again satisfied (now for ${\underline{op}}^k(u_i)$ and ${\overline{op}}^k(u_i)$), the properties (i) and (ii) are also true for these iterations.

• (2) The fact that the functions u and v are strictly increasing (i.e., they represent the corresponding total preorder) does not guarantee that ${\underline{op}}(v)$ and ${\overline{op}}(u)$ are also.

Theorem 4.14

Let $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i=1 }^{n-1} \mathrel {{\mathcal {P}}_{i\, i+1}} )$ a distributed system of n processes with a line communication such that for every $i\in \{1,\ldots ,n-1 \}$ the pair $(u_i,v_i)$ is a representation of the biorder $\overline{{\mathcal {P}}}_{i\,i+1}$, with the additional property that $u_i$ and $v_i$ represent the total orders $\precsim _i$ and $\precsim _{i+1}$, respectively. Then, $(w_1,\ldots ,w_{n})$ is a weak representation of the distributed system, where each function $w_i$ is defined on $X_i$ by a sum of $n-1$ functions as follows:

$w_1$	$=$	$u_1+ \sum _{k=1}^{n-2} {\underline{op}}^k(u_{k+1})$
$w_2$	$=$	$v_1+ u_2+ \sum _{k=1}^{n-3} {\underline{op}}^k(u_{k+2})$
$w_3$	$=$	${\overline{op}}(v_1)+ v_2+u_3+ \sum _{k=1}^{n-4} {\underline{op}}^k(u_{k+3})$
$\vdots$	$\vdots$	$\vdots$
$w_j$	$=$	$\sum _{k=1}^{j-2} {\overline{op}}^{j-1-k}(v_{k}) + v_{j-1} +u_j+ \sum _{k=1}^{n-j-1} {\underline{op}}^k(u_{k+j})$
$\vdots$	$\vdots$	$\vdots$
$w_n$	$=$	$v_{n-1}+ \sum _{k=1}^{n-2} {\overline{op}}^k(v_{n-1-k})$

Proof

First, in the following table, we recover the distinct functions defined on each process:

$X_1$	$X_2$	$X_3$	$\cdots$	$X_{n-1}$	$X_n$
$u_1$	$v_1$	${\overline{op}}(v_1)$	$\cdots$	${\overline{op}}^{n-3}(v_1)$	${\overline{op}}^{n-2}(v_1)$
${\underline{op}}(u_2)$	$u_2$	$v_2$	$\cdots$	${\overline{op}}^{n-4}(v_2)$	${\overline{op}}^{n-3}(v_2)$
${\underline{op}}^2(u_3)$	${\underline{op}}(u_3)$	$u_3$	$\cdots$	${\overline{op}}^{n-5}(v_3)$	${\overline{op}}^{n-4}(v_3)$
$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
${\underline{op}}^{n-2}(u_{n-1})$	${\underline{op}}^{n-3}(u_{n-1})$	${\underline{op}}^{n-4}(u_{n-1})$	$\cdots$	$u_{n-1}$	$v_{n-1}$

Therefore, then, each function $w_i$ is the sum of all these $n-1$ functions defined on the set $X_i$. Let us see now that this tuple $(w_1,\ldots ,w_{n})$ is a weak representation of the distributed system.

Since—by Proposition 4.12 (i)—all the functions defined on each set $X_i$ (for each $i\in \{1,\ldots ,n\}$) are increasing (with respect to the corresponding total order $\precsim _i$) and there is—at least—one which is strictly increasing ($u_i$ and/or $v_{i-1}$), we conclude that the sum of all of them (denoted by $w_i$) is a representation of the total order $\precsim _i$.

Finally, taking into account Proposition 4.12 (ii) and that $(u_i,v_i)$ is a representation of the biorder $\overline{{\mathcal {P}}}_{i\,i+1}$ (for each $i\in \{1, \ldots , n-1\}$) with respect to <, it is straightforward to check that if $x\overline{{\mathcal {P}}}_{i\, i+1} y$ holds then $w_i(x)< w_{i+1}(y)$ is satisfied (for each $i\in \{1, \ldots , n-1\}$ and for any $x\in X_i, \, y\in X_{i+1}$). Therefore, we conclude that $(w_1,\ldots ,w_{n})$ is a weak representation of the distributed system. $\square$

5 Quasi-finite Partial Orders

In this section, a particular but interesting class of partial orders is studied: quasi-finite partial orders. This kind of structures includes all those partial orders that can be understood as a finite family of chains with a communication. The key of this section is to focus the research on the quotient sets (with respect to the traces of the biorders), which makes possible a discrete study of the representability, achieving results not only of the quotient structure but also of the original one. Thus, it is also possible to apply some techniques on finite posets as those introduced in [19].

In fact, given a distributed system $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$, we may define an equivalence relation ${\mathcal {I}}_i$ on $X_i$ by means of the intersection of all the equivalence relations ${\mathcal {I}}_{ij}^*$ and ${\mathcal {I}}_{ji}^{**}$ (for any $i\ne j$) on $X_i$ (i.e., the equivalence relation associated to the union of all the traces on $X_i$) (see Remark 2.7). Then, since the communication is a finite relation, the cardinal of each quotient set ${\overline{X}}_i=X_i /{\mathcal {I}}_i$ is finite, for any $i=1, \ldots , n$.

The goal of the present section is the attainment of a method to construct finite Richter–Peleg multi-utility representations for quasi-finite partial orders, i.e., a representation method for distributed systems. For more clarity, Example 5.4 is included in order to show this procedure.

Let us see how quasi-finite partial orders are defined.

Definition 5.1

We shall say that a partial order on a set is quasi-finite if it is the causal relation of a distributed system.

Remark 5.2

• (1) By definition, quasi-finite partial orders are near-complete.

• (2) Given a distributed system $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$, we may be interested just in the communication $\mathrel {{\mathcal {P}}}$ and skip the remaining information, i.e., the total orders $\precsim _i$. In that case, a finite poset $(\bigcup _{i=1}^n (X_i/{\mathcal {I}}_i, {\overline{\precsim }}_i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$ is achieved, where ${\overline{\precsim }}_i$ is the total order on ${\overline{X}}_i=X_i/{\mathcal {I}}_i$ and now the communication $\mathrel {{\mathcal {P}}}$ is restricted to the quotient sets (see Example 11 in [16]).

The following proposition shows how to construct a Richter–Peleg multi-utility representation of a quasi-finite partial order, just starting from a bijective Richter–Peleg multi-utility representation (see Definition 2.9) of a finite poset and utilities of total preorders.

Theorem 5.3

Let $\precsim$ be a quasi-finite partial order on X that coincides with the causal relation associated to a distributed system $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$. Let $\{w_i\}_{i=1}^n$ be a family of utilities⁷$w_i:(X_i, \precsim _i)\rightarrow (0,1)$ and ${\mathcal {U}}=\{u_l\}_{l=1}^k$ a bijective Richter–Peleg multi-utility representation associated to $(\bigcup _{i=1}^n (X_i/{\mathcal {I}}_i, {\overline{\precsim }}_i),$ $\mathrel {{\mathcal {P}}} )$. Then, the family of functions ${\mathcal {V}}=\{v_l\}_{l=1}^k$ defined by

$$\begin{aligned} v_l(x)= u_l({\bar{x}})+ w_i(x), \quad \text {if } x\in X_i, \forall x\in X\end{aligned}$$

is a Richter–Peleg multi-utility representation of the quasi-finite partial order $\precsim$.

Proof

$\Rightarrow :$ Let x, y be two elements in X such that $x\prec y$.

If x and y belong to the same process $X_i$, then $x\prec _i y$ so, $w_i(x)<w_i(y)$. Since $x\prec _i y$, it is also true that $x\precsim _{ij}^* y$ as well as $x\precsim _{ji}^{**} y$ for any trace defined on $X_i$. Therefore, $u_l({\bar{x}})\le u_l({\bar{y}})$ is satisfied, for any function $u_l\in {\mathcal {U}}$. Hence, we conclude that $v_l(x)<v_l(y)$ for any $v_l\in {\mathcal {V}}$.

If x and y belong to distinct processes, $X_i$ and $X_j$ respectively, then it holds that $x\overline{{\mathcal {P}}}_{ij} y$ so, $u_l({\bar{x}})<u_l({\bar{y}})$ is satisfied for any $u_l\in {\mathcal {U}}$. Therefore, since $u_l({\bar{x}})+1\le u_l({\bar{y}})$ and the codomain of the utilities is the interval (0, 1), we conclude that $v_l(x)<v_l(y)$ for any $v_j\in {\mathcal {V}}$.

$\Leftarrow :$ Suppose that $v_l(x)<v_l(y)$ is satisfied for any $v_l\in {\mathcal {V}}$. Since $v_l(x)= u_l({\bar{x}})+ w_i(x)$ (with $x\in X_i$, for any $i=1,\ldots , n$) and the codomain of the utilities is the interval (0, 1), $v_l(x)<v_l(y)$ holds true if and only if $u_l({\bar{x}})\le u_l({\bar{y}})$ for any $u_l\in {\mathcal {U}}$.

Now, we distinguish two cases.⁸

• 1. $u_l({\bar{x}})< u_l({\bar{y}})$ for any $u_l\in {\mathcal {U}}$. In that case, we distinguish again two cases.

  • (a) x and y belong to the same process $X_i$. In that case, since $u_l({\bar{x}})< u_l({\bar{y}})$ for any $u_l\in {\mathcal {U}}$, there exists a trace $\precsim _{ij}^*$ or $\precsim _{ji}^{**}$ on $X_i$ such that $x\prec _{ij}^* y$ or $x\prec _{ji}^{**} y$. Thus, we conclude that $x\prec _i y$ and, hence, $x\prec y$.

  • (b) x and y belong to distinct processes, $X_i$ and $X_j$ respectively. In that case, since $u_l({\bar{x}})< u_l({\bar{y}})$ for any $u_l\in {\mathcal {U}}$, there exist elements $x_{k_1}\in X_{k_1},\ldots ,x_{k_s}\in X_{k_s}$ (for some $k_1, \ldots ,k_s\in \{1,\ldots ,n\}$) such that $x\mathrel {\overline{{\mathcal {P}}}}_{i\,k_1} x_{k_1} \mathrel {\overline{{\mathcal {P}}}}_{k_1\,k_2}\cdots \mathrel {\overline{{\mathcal {P}}}}_{k_{s-1}\, k_s} x_k \mathrel {\overline{{\mathcal {P}}}}_{k_s\,j} y$. Thus, we conclude that $x\prec y$.

• 2. $u_l({\bar{x}})= u_l({\bar{y}})$ for any $u_l\in {\mathcal {U}}$. In that case, x and y belong to the same quotient class and, therefore, to the same process. Thus, since $u_l({\bar{x}})= u_l({\bar{y}})$ for any $u_l\in {\mathcal {U}}$ and $v_l(x)<v_l(y)$ for any $v_l\in V$, it holds that $w_i(x)<w_i(y)$. Then, we conclude that $x\prec _i y$ and, hence, $x\prec y$.$\square$

Fig. 9

The distributed system of three processes of Example 5.4

Example 5.4

Let $(X_1, \precsim _1), (X_2, \precsim _2)$ and $(X_3, \precsim _3)$ be three totally ordered sets (they may be uncountable, but representable in any case) such that $x_i= \max \{(X_i, \precsim _i)\}$ and $y_i= \min \{(X_i, \precsim _i)\}$, for $i=1,2,3.$ Suppose that there is a communication between these sets (defined by ${\mathcal {P}}_{12},{\mathcal {P}}_{13}$ and ${\mathcal {P}}_{32}$) as it is shown in Fig. 9, such that $y_1\mathrel {{\mathcal {P}}}_{13} z_3,$ $x_1\mathrel {{\mathcal {P}}}_{12} y_2$ and $x_3\mathrel {{\mathcal {P}}}_{32} y_2$.

If we focus on the quotient we achieve the poset of Fig. 10. Here, it is straightforward to check that ${\overline{x}}_1=U_{\prec _1}(y_1)\subseteq X_1$, ${\overline{y}}_1=L_{\precsim _1}(y_1)\subseteq X_1$, ${\overline{x}}_2=X_2$, ${\overline{x}}_3=U_{\precsim _3}(z_3)\subseteq X_3$ and ${\overline{y}}_3=L_{\prec _3}(z_3)\subseteq X_3$.

Fig. 10

Hasse diagram and the corresponding bijective Richter–Peleg multi-utility $\{u_1,u_2\}$ of the quotient associated to the distributed system of Fig. 9

Therefore, now, given $w_1, w_2$ and $w_3$ three representations (that take values on (0, 1)) of $\precsim _1$, $\precsim _2$ and $\precsim _3$, respectively, we can easily construct a representation of the distributed system through these functions and the utilities of the poset, as commented in Theorem 5.3:

$$\begin{aligned}{} & {} v_1(x)= u_1({\bar{x}})+ w_i(x), \quad \text {if } x\in X_i, \forall x\in X, \\ {}{} & {} v_2(x)= u_2({\bar{x}})+ w_i(x), \quad \text {if } x\in X_i, \forall x\in X. \end{aligned}$$

It is straightforward to see that $\{v_1,v_2\}$ is also a Richter–Peleg multi-utility of the causal relation (see Proposition 4.4).

In the theorem before, the functions of the representation are defined through the sum of two functions. Hence, it is possible to study the continuity of the functions of the representation by means of the continuity of the other ones.

Theorem 5.5

Let $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}}=\bigcup _{i\ne j} \mathrel {{\mathcal {P}}_{ij}} )$ be a distributed system where each set $X_i$ is endowed with a topology $\tau _i$. Let ${\mathcal {I}}_i$ be the equivalence relation on $X_i$ emerged from the intersection of all the equivalence relations $\sim _{ij}^*$ and $\sim _{ji}^{**}$ (for any $i\ne j$) on $X_i$. Assume that the following conditions are satisfied for each $i=1,\ldots ,n$:

• (i) The total orders $\precsim _i$ are $\tau _i$-continuous and representable.

• (ii) Each class ${\overline{x}}=\{y\in X_i\, :\, y{\mathcal {I}}_i x\}$ is open in $X_i$, for any $x\in X_i$.

Then, the distributed system is continuously representable.

Proof

By Theorem 5.3, we may construct a representation of the distributed system such that each function $v_l$ is defined in $X_i$ by $v_l(x)= u_l({\bar{x}})+ w_i(x)$ (as stated in Theorem 5.3). By hypothesis (i), we may assume that $w_i$ is continuous [9], and by condition (ii) it is straightforward to see that $u_l$ is continuous too. Hence, each function $v_l$ is continuous in $X_i$, for any $i=1,\ldots ,n$. $\square$

Remark 5.6

The reciprocal of the theorem above is not true (see Example 11 and Remark 12 of [16]).

From Theorem 5.5 and Proposition 4.4 the following corollary is deduced, which may be useful if we are focusing on a quasi-finite partially ordered set $(X,\precsim )$ endowed with a topology.

Corollary 5.7

Let $\precsim$ be a quasi-finite partial order on $(X,\tau )$. Assume that there exists a distributed system $(\bigcup _{i=1}^n (X_i, \precsim _i), \mathrel {{\mathcal {P}}} )$ such that the following conditions are satisfied for each $i=1,\ldots ,n$:

• (i) The total orders $\precsim _i$ are $\tau _i$-continuous and representable, where $\tau _i=\tau _{|_{X_i}}$.

• (ii) Each class ${\overline{x}}=\{y\in X_i\, :\, y{\mathcal {I}}_i x\}$ is open in $X_i$, for any $x\in X_i$.

• (iii) Any open set $U\in \tau _i$ is also open in $\tau$.

Then, there exists a continuous and finite Richter–Peleg multi-utility of the partial order $\precsim$ on $(X,\tau )$.

6 Further Comments

For a sake of brevity and clearness, in the present paper, we have argued on total orders and partial orders, however, it can be easily generalized to total preorders and preorders.

The sections related to representability (and the aggregation problem) may be implemented through partial functions, using the idea of partial representability (see [7]). In order to illustrate this final idea, we include the following result:

Proposition 6.1

Let $(X_1,\precsim _1), (X_2,\precsim _2)$ and $(X_3,\precsim _3)$ be three representable totally ordered sets and ${\mathcal {P}}_{12}$ and ${\mathcal {P}}_{23}$ two communications from $X_1$ to $X_2$ and from $X_2$ to $X_3$, respectively. Thus, the structure that arise is a distributed system of three processes with line communication. Assume that each biorder is representable, such that:

$$\begin{aligned} x\overline{{\mathcal {P}}}_{12} y \iff u_1(x)<v_1(y), \text { for any } x\in X_1, y\in X_2,\\ y\overline{{\mathcal {P}}}_{23} z \iff v_2(y)<w_1(z), \text { for any } y\in X_2, z\in X_3, \end{aligned}$$

as well as the biorder $\overline{{\mathcal {P}}}_{13}$ emerged from the composition $\overline{{\mathcal {P}}}_{12}\circ \overline{{\mathcal {P}}}_{23}$ (i.e., $x \overline{{\mathcal {P}}}_{13} z \iff x\overline{{\mathcal {P}}}_{12} y \overline{{\mathcal {P}}}_{23} z$, for some $y\in X_2$) is representable by $(u_2,w_2):$

$$\begin{aligned} x\overline{{\mathcal {P}}}_{13} z \iff u_2(x)<w_2(z), \text { for any } x\in X_1, z\in X_3.\end{aligned}$$

(Here, we assume that the functions $u_1,u_2,v_1, v_2, w_1$ and $w_2$ takes values on (0, 1), as well as they also represent the total order of the corresponding set). Then, the associated causal relation $\rightarrow$ is partially representable (see [7]) through the functions $\{\sigma _1, \sigma _2, \sigma _3\}$ defined as follows:

$$\begin{aligned} \sigma _1(x)=\left\{ \begin{array}{ll} u_1(x) &{}{; } x\in X_1 \\ v_1(x) &{}{; } x\in X_2 \\ w_1(x)+1 &{}\text{; } x\in X_3 \\ \end{array}\right. \qquad \sigma _2(x)=\left\{ \begin{array}{ll} u_1(x) &{}\text{; } x\in X_1 \\ v_2(x)+1 &{}\text{; } x\in X_2 \\ w_1(x)+1 &{}\text{; } x\in X_3 \\ \end{array}\right. \end{aligned}$$

$$\begin{aligned} \sigma _3(x)=\left\{ \begin{array}{ll} u_2(x) &{}\text{; } x\in X_1 \\ \emptyset &{}\text{; } x\in X_2 \\ w_2(x) &{}\text{; } x\in X_3 \\ \end{array}\right. \end{aligned}$$

So that, $\quad x\rightarrow y$ if and only if $\sigma (x)<\sigma (y)$ for some $\sigma \in \{\sigma _i\}_{i=1}^3$ as well as $\sigma _i(x)<\sigma _i(y)$ for any $i=1,2,3$ such that $\sigma _i$ is defined on both x and y.