Foundations of Physics

, Volume 44, Issue 7, pp 792–811 | Cite as

How to Produce S-Tense Operators on Lattice Effect Algebras

Article

Abstract

Tense operators in effect algebras play a key role for the representation of the dynamics of formally described physical systems. For this, it is important to know how to construct them on a given effect algebra \( E\) and how to compute all possible pairs of tense operators on \( E\). However, we firstly need to derive a time frame which enables these constructions and computations. Hence, we usually apply a suitable set of states of the effect algebra \( E\) in question. To approximate physical reality in quantum mechanics, we use only the so-called Jauch–Piron states on \( E\) in our paper. To realize our constructions, we are restricted on lattice effect algebras only.

Keywords

Effect algebra MV-algebra Complete lattice Tense operator S-tense operator Jauch–Piron E-state Jauch–Piron E-semi-state 

Mathematics Subject Classification

Primary 03B44 03G25 06A11 06B23 

1 Introduction

Logic of quantum mechanics is an important tool for deciding and evaluation of propositions and propositional formulas on a physical system describing events in microcosmos. As known in quantum physics, behaviour of these physical systems and their elements (i.e., elementary particles) differs from physical systems which are observed in classical physics and whose behaviour is ruled by the classical propositional calculus. This was the reason that Foulis and Bennett [15] introduced the so-called effect algebras describing algebraic properties of propositions on events in quantum mechanics. For a more detailed motivation, the reader is referred to the monograph [12] by Dvurečenskij and Pulmannová.

Every physical system \(\mathcal P\) is a dynamic one which means that the true values of propositions about \(\mathcal P\) vary in time. However, the majority of propositional calculi used for a description of \(\mathcal P\) traditionally do not incorporate the time dimension. This means that the propositions are considered relative to a given moment. On the other hand, for a physical system \(\mathcal P\) there are usually discerned its states.

We assume that \(\mathcal P\) is in a state \(s_1\) at time \(t_1\) and it goes to a state \(s_2\) at time \(t_2\) (as an example may serve Minkowski spacetime \(\mathbb {R}^{1,3}\) where three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime—elements of \(\mathbb {R}^{1,3}\) are then states in our setting). Hence this transition from the state \(s_1\) to \(s_2\) can be taken as a movement from time \(t_1\) to \(t_2\) in the time scale induced by the set \(S\) of states. Since not from every state \(s_1\) we can realize a transition to an arbitrary state of \(S\), there is a non-trivial binary relation \(R\) on \(S\) such that \((s_1,s_2) \in R\) if and only if the system \(\mathcal P\) can switch from \(s_1\) to \(s_2\). Hence, we take advantage to think of the set \(S\) of states as of the time scale and of the relation \(R\) as time preference, i.e., \((s_1,s_2) \in R\) if \(s_1\) is “before” \(s_2\) or \(s_2\) is “after” \(s_1\). Assuming that \((s_1,s_2) \in R\), \((s_2,s_3) \in R\) and possibly \(s_1=s_3\) means, among other things, that time loops are allowed. The couple \((S,R)\) is then the so-called time frame. Having a time frame of a physical system \(\mathcal P\), we can quantify our propositions in time as follows.

Let \(p(t)\) be a propositional formula of a logic of the system \(\mathcal P\) having only one free variable \(t\) which plays the role of time. We introduce two of the so-called tense operators \(G\) and \(H\) which quantify \(p(t)\) over time as follows:
  • \(G(p)(s)\) is valid if for any \(t\) with \((s, t)\in R\) the formula \(p(t)\) is valid,

  • \(H(p)(s)\) is valid if for any \(t\) with \((t, s)\in R\) the formula \(p(t)\) is valid.

Thus the unary operator \(G\) is a tense operator saying “it is always going to be the case that” and \(H\) is saying “it has always been the case that”.
Our first question concerns an algebraic axiomatization of tense operators. It was already studied for effect algebras in [6, 7], for Boolean algebras in [3], for MV-algebras in [2, 10] and for de Morgan algebras in [8, 14]. This axiomatization will be used also here. However, having an effect algebra \(E\) as an axiomatization of the propositional logic of a given physical system \(\mathcal P\), and the tense operators \(G\) and \(H\) on \( E\), we can ask about the following:
  1. (a)

    if a time frame is given (or it is already constructed), how can we compute tense operators \(G^{*}\) and \(H^{*}\) on \( E\) having at least the same values for propositional formulas \(p(t)\) as by the given operators \(G\) and \(H\).

     
  2. (b)

    how to construct all possible pairs of tense operators on \( E\).

     
  3. (c)

    having tense operators satisfying our axiomatization on the lattice effect algebra \(E\) only, i.e., if a time frame is not given, under which conditions on \(E\) we can create the time frame in order to enable the solution of the above mentioned questions (a) and (b). This is usually called a representation problem.

     
The representation problem goes back to Rutledge (see [21]) where he studied monadic MV-algebras as an algebraic model for the monadic predicate calculus of Łukasiewicz infinite-valued logic, in which only a single individual variable occurs. Recall that monadic MV-algebras are a particular case of tense MV-algebra (see [10]). Rutledge represented each subdirectly irreducible monadic MV-algebra as a subalgebra of a functional monadic MV-algebra.

In fact, some solutions of above questions were already found by the authors, see e.g. [7, 16] and [17]. However, the next question is how our formal and purely algebraic approach coresponds to a real physical quantum system \(\mathcal P\).

For this sake, we prefer not to use all possible states of \(\mathcal P\) but only the so-called Jauch–Piron states. The advantage is that these states reflect an important property of the logic of quantum mechanics, namely the so-called Jauch–Piron property saying that if the probability of propositions \(p\) and \(q\) being true is zero then there exits a proposition \(r\) covering both \(p\) and \(q\) such that the probability of \(r\) being true is also zero. In our paper, we start with lattice effect algebras for the sake of simplicity. Hence, we can take \(r = p \vee q\). This is in accordance with Kolmogorovian probability theory. The study of Jauch–Piron states was motivated by the requirement that states on projection structures that qualify for having physical meaning should satisfy the natural Jauch–Piron property (see [18]).

The paper is organized as follows. In Sect. 2, we develop and study the basic concepts of tense effect algebras and an explanation of the role of time scale. We also give a justification for considering S-tense operators and present the main results of the paper which are the following:
  1. (a)

    any time frame \((T, R) \) on a linearly ordered complete MV-algebra \(M \) induces S-tense operators on the cartesian product \(M^{T}\),

     
  2. (b)

    conversely, any S-tense lattice effect algebra with a suitable set \(T\) of Jauch–Piron states is representable by some time frame \((T,R)\) as a subalgebra of the respective cartesian product \([0,1]^{T}\) equipped with the respective S-tense operators from (a).

     
The main results are based on Sects. 3, 4 and 5. In Sect. 3 we start with the notion of an (Jauch–Piron) E-semi-state on a lattice effect algebra motivated by the fact that a composition of an E-tense operator with an (Jauch–Piron) E-state is an (Jauch–Piron) E-semi-state. The crucial result of this section is Proposition 11 that any Jauch–Piron E-semi-state on an E-representable lattice effect algebra \(E\) is a meet of E-states.
In Sect. 4 we introduce the notion of a (very) Jauch–Piron EM-function between lattice effect algebras which generalizes the notion of E-tense operator. Also we generalize the notion of time frame by a notion of frame. The main result of this section is the following:
  1. (c)

    any frame \((S, T, R) \) on a linearly ordered complete MV-algebra \(M \) induces a very Jauch–Piron EM-function from \(M^{T}\) into \(M^{S}\).

     
In Sect. 5 we solve the representation problem for very Jauch–Piron EM-functions beween lattice effect algebras. We also give an example of an E-representable E-Jauch–Piron lattice effect algebra which is not an MV-algebra and we list all tense, E-tense and S-tense operators on it.

2 S-Tense Operators on Lattice Effect Algebras

In this section we develop and study the basic concepts of tense effect algebras and an explanation of the role of time scale. We also give a justication for considering S-tense operators, describe their construction and a framework for their representation.

By an effect algebra is meant a structure \((E;+,0,1)\) where \(0\) and \(1\) are distinguished elements of \(E\), \(0\ne 1\), and \(+\) is a partial binary operation on \(E\) satisfying the following axioms for \(p,q,r\in E\):
  1. (E1)

    if \(p+q\) is defined then \(q+p\) is defined and \(p+q=q+p\)

     
  2. (E2)

    if \(q+r\) is defined and \(p+(q+r)\) is defined then \(p+q\) and \((p+q)+r\) are defined and \(p+(q+r)=(p+q)+r\)

     
  3. (E3)

    for each \(p\in E\) there exists a unique \(p'\in E\) such that \(p+p'=1\); \(p'\) is called a supplement of \(p\)

     
  4. (E4)

    if \(p+1\) is defined then \(p=0\).

     
We often denote the effect algebra \((E;\oplus ,0,1)\) briefly by \(E\). In every effect algebra \(E\) we can introduce the induced order\(\le \) on \(E\) and the partial operation \(-\) as follows
$$\begin{aligned} x\le y\quad \text { if for some }z\in E\quad x+z=y, z=y-x \end{aligned}$$
(see e.g. [12] for details). Then \((E;\le )\) is an ordered set and \(0\le x\le 1\) for each \(x\in E\).

It is worth noticing that \(a+b\) exists in an effect algebra \({E}\) if and only if \(a\le b'\) (or equivalently, \(b\le a'\)). This condition is usually expressed by the notation \(a\bot b\) (we say that \(a,b\) are orthogonal). Dually, we have a partial operation \(\cdot \) on \(E\) such that \(a\cdot b\) exists in an effect algebra \({E}\) if and only if \(a'\le b\) in which case \(a\cdot b=(a'+b')'\). This allows us to equip \(E\) with a dual effect algebraic operation such that \({E}^{op}=(E;\cdot ,1,0)\) is again an effect algebra, \('^{{E}^{op}}='^{{E}}='\) and \(\le _{{E}^{op}}=\le ^{op}\).

A morphism of effect algebras is a map between them such that it preserves the partial operation \(+\), the bottom and the top elements. A map \(s:E\rightarrow [0,1]\) is called a state on \({E}\) if \(s(0)=0\), \(s(1)=1\) and \(s(x+ y)=s(x)+s(y)\) whenever \(x+ y\) exists in \({E}\). Note that the real unit interval \([0,1]\) is an effect algebra such that \(x+ y\) exists in \([0,1]\) if and only if the usual sum of \(x\) and \(y\) is less or equal to \(1\).

A morphism\(f:P_1\rightarrow P_2\)of bounded posets is an order, top element and bottom element preserving map. Any morphism of effect algebras is a morphism of corresponding bounded posets. A morphism \(f:P_1\rightarrow P_2\) of bounded posets is order reflecting if, for all \(a, b\in P_1\),
$$\begin{aligned} f(a)\le f(b) \ \hbox {if and only if} \ a\le b. \end{aligned}$$
An isomorphism of effect algebras is a surjective order reflecting morphism of effect algebras. In particular, \(':{{E}}\rightarrow {{E}^{op}}\) is an isomorphism of effect algebras.

If, moreover \((E;\le )\) is a lattice (with respect to the induced order), then \({E}\) is called a lattice effect algebra. On any lattice effect algebra \({E}\) we may introduce total operations \(\oplus \) and \(\odot \) as follows: \(x\oplus y=x+(y\wedge x')\) and \(x\odot y= (x'\oplus y')'\).

Given a positive integer \(n\in {\mathbb N}\), we let \(n\times x= x \oplus x \oplus x \cdots \oplus x\), \(n\) times, \(x^{n}= x \odot x \odot x \cdots \odot x\), \(n\) times, \(0x = 0\) and \(x^{0} = 1\).

Note that a lattice effect algebra \({E}\) is an MV-algebra (see [9]) with respect to the operations \(\oplus \) and \('\) if and only if \(x\wedge y=0\) implies \(x\le y'\). In this case are \(\oplus \) and \(\odot \) order preserving operations. It follows that any linearly ordered effect algebra is an MV-algebra.

An E-morphism between lattice effect algebras is a morphism \(f:{{E}}_1\rightarrow {{E}}_2\) of effect algebras such that
  1. (i)

    \(f(x\oplus x)=f(x)\oplus f(x)\).

     
Note that from the condition (i) we automatically have the following condition:
  1. (ii)

    \(f(x\odot x)=f(x)\odot f(x)\).

     
Any isomorphism of lattice effect algebras is an E-morphism which preserves lattice operations as well.

The axiomatization of tense operators \(G\) and \(H\) in the classical propositional logic is given in [3]. For effect algebras, it was settled in [6, 7] and [16]. We can repeat the definition.

Let \((E;+,0,1)\) be an effect algebra. Unary operators \(G\) and \(H\) on \({E}\) are called partial tense operators if they are partial mappings of \(E\) into itself satisfying the following axioms:
  1. (T1)

    \(G(0)=0\), \(G(1)=1\), \(H(0)=0\) and \(H(1)=1\),

     
  2. (T2)

    \(x\le y\) implies \(G(x)\le G(y)\) whenever \(G(x), G(y)\) exist and \(H(x)\le H(y)\) whenever \(H(x), H(y)\) exist,

     
  3. (T3)

    if \(x+y\) and \(G(x), G(y), G(x+y)\) exist then \(G(x)+G(y)\) exists and \(G(x)+G(y)\le G(x+y)\) and if \(x+y\) and \(H(x), H(y), H(x+y)\) exist then \(H(x)+H(y)\) exists and \(H(x)+H(y)\le H(x+y)\),

     
  4. (T4)

    \(x\le GP(x)\) if \(H(x')\) exists, \(P(x)=H(x')'\) and \(GP(x)\) exists, \(x\le HF(x)\) if \(G(x')\) exists, \(F(x)=G(x')'\) and \(HF(x)\) exists.

     
If both \(G\) and \(H\) are total (i.e., \(G\) and \(H\) are mappings of \(E\) into itself defined for each \(x\in E\)) then \(G\) and \(H\) are called tense operators and the triple \(({E};G,H)\) is called a tense effect algebra.

This is exactly the case when a time frame \((T, R)\) is not explicitly mentioned. But it may happen, if a time scale \(T\) and a relation \(R\) for this axiomatization were constructed, that the obtained binary relation \(R\) is neither reflexive nor transitive. The conditions under which \(R\) will be a quasi-order are analysed in Theorems 1 and 3.

One can immediately mention that if \((E;+,0,1)\) is an effect algebra and \(G\) and \(H\) are mappings of \(E\) into itself defined by \(G(1)=1=H(1)\) and \(G(x)=0=H(x)\) for all \(x\in E\), \(x\not = 1\) then \(G\) and \(H\) are tense operators. However, these operators reveal little about the physical system \(\mathcal {P}\) because all that is not identically equal to \(1\) is considered to be false both in the past and in the future. Hence, we are searching to find tense operators on \({E}\) having maximally many values. The construction of an important class of such operators is given below in Theorem 15. However, it can happen that tense operators with maximally many values can loose their physical interpretation but another couple with smaller values can be more appropriate.

Since any lattice effect algebra can be covered by its maximal sub-MV-algebras called blocks [19] it is quite natural to ask that our (total) tense operators behave on MV-algebras according to the axiomatization of tense MV-algebras given by Diaconescu and Georgescu in [10]. This can be accomplished by the following axioms formulated for lattice effect algebras:
  1. (T5)

    \(G(x\oplus x)=G(x)\oplus G(x), H(x\oplus x)=H(x)\oplus H(x)\),

     
  2. (T6)

    \(G(x\odot x)=G(x)\odot G(x), H(x\odot x)=H(x)\odot H(x)\).

     
We call such tense operators \(G\) and \(H\)  E-tense operators. Note that an embedding between tense effect algebras is an order reflecting morphism of effect algebras such that it commutes with the corresponding tense operators.

Let us denote by \(S(E)=\{x\in E \mid x\wedge x'=0\}\) the set of all sharp elements of \(E\). Then, for any pair \(G\) and \(H\) of E-tense operators on \(E\) and any \(x\in S(E)\), we have that \(G(x)\) and \(H(x)\) are in \(S(E)\).

Now, in our case, by a time frame is meant a couple \((T,R)\) where \(T\) is a non-void set and \(R\subseteq T\times T\) is a reflexive and transitive relation. The last condition on \(R\) comes from the interpretion of \((s, t) \in R\) (or equivalently \(s R t\)) as \(s\) is before \(t\) in the non-strict sense so it would seem that \(R\) should be at least a quasi-order.

Having a lattice effect algebra \((E;+,0,1)\) and a non-void set \(T\), we can produce the direct power \((E^T;+, o,j)\) where the operation \(+\) and the induced operations \(\vee \), \(\wedge \), \(\oplus \), \(\odot \) and \(\lnot \) are defined and evaluated on \(x,y\in E^T\) componentwise. Moreover, \(o, j\) are such elements of \(E^T\) that \(o(t)=0\) and \(j(t)=1\) for all \(t\in T\). The direct power \({E}^T\) is again a lattice effect algebra.

The following theorem is an immediate corollary of Theorem 15 so we shall omit the proof up to Sect. 4.

Theorem 1

Let \({M}\) be a linearly ordered complete MV-algebra, \((T, R)\) be a time frame and \(G, H\) be maps from \({M}^T\) into \({M}^T\) defined by
$$\begin{aligned} G(x)(s) = \bigwedge \{x(t) \mid t \in T, sRt\} \end{aligned}$$
and
$$\begin{aligned} H(x)(s) = \bigwedge \{x(t) \mid t \in T, tRs\} \end{aligned}$$
for all \(x\in M^T\) and \(s\in T\). Then \(G\) and \(H\) are E-tense operators on the lattice effect algebra \({M}^T\) such that
  1. (T7)

    \(G(x)\le x, H(x)\le x\),

     
  2. (T8)

    \(G(x)=G(G(x)), H(x)=H(H(x))\).

     

Motivated by Theorem 1 we say that E-tense operators \(G\) and \(H\) on a lattice effect algebra \(E\) are S-tense whenever they satisfy axioms (T7) and (T8) and that \(E\) is an S-tense lattice effect algebra.

It follows that \(G(x)\) means that “\(x\) is always going to be the case – starting now” and \(H(x)\) means that “\(x\) was always, and is now the case”. Moreover, the condition (T8) yields that the statement “\(x\) will always be the case” is equivalent to the statement “it will always be the case that \(x\) will always be the case”.

A functional S-tense lattice effect algebra\(E\) is an S-tense lattice effect algebra with S-tense operators \(G_E\) and \(H_E\) which is an E-effect-algebraic reduct of \([0,1]^{T}\) for a time frame \((T, R)\) such that \(G_E\) is a restriction of \(G\) from Theorem 1 on \(E\) and \(H_E\) is a restriction of \(H\) from Theorem 1, respectively.

By a functional representation of an S-tense lattice effect algebra\(A\) with S-tense operators \(G_A\) and \(H_A\), we mean simply a functional S-tense lattice effect algebra \(E\) such that there is an isomorphism of effect algebras \(f:A \rightarrow E\) satisfying \(f\circ G_A=G_E \circ f\) and \(f\circ H_A=H_E \circ f\).

As mentioned in Introduction, the set of states of an effect algebra can serve as a time scale over which we can quantify propositions on a physical system \({\mathcal P}\). Hence, it is worth to know as much as possible about states on a given effect algebra. For this, we introduce particular states, named E-states and Jauch–Piron E-states, which can be used for a representation of an effect algebra into a direct product of canonical effect algebras on the interval \([0,1]\). Although it can be introduced in full generality, we are concentrating only on the case of lattice effect algebras in order to reach everywhere defined mappings.

Definition 2

Let \((E;+,0,1)\) be a lattice effect algebra. A map \(s:{E} \rightarrow [0,1]\) is called
  1. (1)

    an E-state on\({E}\) if \(s\) is an E-morphism of effect algebras;

     
  2. (2)

    a Jauch–Piron E-state on\({E}\) if \(s\) is an E-state and

     
  1. (JP)

    \(s(x)=1= s(y)\) implies \(s(x\wedge y)=1\).

     
Let \((E;+,0,1)\) be a lattice effect algebra.
  1. (3)

    If there exists an order reflecting set \(T\) of E-states (Jauch–Piron E-states, respectively) on \({E}\) then \({E}\) is said to be E-representable (E-Jauch–Piron representable, respectively).

     
  2. (4)

    If any E-state is E-Jauch–Piron then \({E}\) is called an E-Jauch–Piron lattice effect algebra.

     

First, note that any E-Jauch–Piron lattice effect algebra with an order reflecting set of E-states is E-Jauch–Piron representable. Also, any E-Jauch–Piron representable lattice effect algebra is E-representable.

Second, if \({E}\) is E-representable then the induced morphism \({i_{{E}}^T}:{{E}} \rightarrow [0,1]^{T}\) (sometimes called an embedding) is an order reflecting morphism of effect algebras such that \({i_{{E}}^T}(x\oplus x)={i_{{E}}^T}(x)\oplus {i_{{E}}^T}(x)\) and \({i_{{E}}^T}(x\odot x)={i_{{E}}^T}(x)\odot {i_{{E}}^T}(x)\) for all \(x\in E\).

Third, if there exists an order reflecting set \(T\) of states that are also lattice morphisms then \(E\) is an MV-algebra that is E-Jauch–Piron representable.

Fourth, if \({E}\) is an MV-algebra and \(s\) a state on \({E}\) then we always have \(s(x\vee y)+s(x\wedge y)=s(x)+s(y)\) for all \(x, y\in E\). Hence in every MV-algebra any state \(s\) satisfies (JP). Moreover from [2] we know that \(s\) is an E-state if and only if \(s\) is an extremal state (MV-algebra morphism).

The next theorem gives us a solution of the representation problem for S-tense operators. It immediately follows from Theorem 19 so we postpone its proof until Sect. 5.

Theorem 3

Let \( E\) be an E-representable E-Jauch–Piron S-tense lattice effect algebra with S-tense operators \(G_E\) and \(H_E\). Then \((E,G_E,H_E)\) can be embedded into the tense MV-algebra \(([0,1]^T,G,H)\) induced by time frame \((T,\rho _G),\) where \(T\) is the set of all Jauch–Piron E-states from \( E\) to \([0,1]\) and the relation \(\rho _G\) is defined by
$$\begin{aligned} s\rho _G t \hbox { if and only if } s(G_E(x)) \le t(x)\hbox { for any } x\in E. \end{aligned}$$
In particular, the following diagram of functions commutes:

3 E-Semi-States on Lattice Effect Algebras

In this section we will systematically study the notion of an (Jauch–Piron) E-semi-state on a lattice effect algebra. The study of (Jauch–Piron) E-semi-states was motivated by the fact that a composition of an E-tense operator with an (Jauch–Piron) E-state is an (Jauch–Piron) E-semi-state. The basic result of this section is Proposition 11 that any Jauch–Piron E-semi-state on an E-representable lattice effect algebra \(E\) is a meet of E-states.

Definition 4

Let \((E;+,0,1)\) be a lattice effect algebra. A map \(s:{E} \rightarrow [0,1]\) is called
  1. (1)
    an E-semi-state on\({E}\) if
    1. (i)

      \(s(0)=0, s(1)=1,\)

       
    2. (ii)

      \(s(x)+ s(y)\le s (x+ y)\) whenever \(x+ y\) is defined,

       
    3. (iii)

      \(s(x)\odot s(x)=s(x\odot x),\)

       
    4. (iv)

      \(s(x)\oplus s(x)=s(x\oplus x),\)

       
     
  2. (2)
    a Jauch–Piron E-semi-state on\({E}\) if \(s\) is an E-semi-state and
    1. (JP)

      \(s(x)=1= s(y)\) implies \(s(x\wedge y)=1\).

       
     

Lemma 5

Let \((E;+,0,1)\) be a lattice effect algebra, \(s:{E} \rightarrow [0,1]\) a Jauch–Piron E-semi-state on \({E}\). Then \(s\) satisfies the following condition:
$$\begin{aligned} s(x)=1= s(y) \hbox { and } x\cdot y \hbox { defined implies } s (x\cdot y)=1. \end{aligned}$$

Proof

Assume that \(s(x)=1= s(y)\) and \(x\cdot y\) is defined. Then there is a block \(M\) (see [12]) of \({E}\) which is an MV-algebra containg \(x\) and \(y\). Hence also \(x\wedge y, x\odot y=x\cdot y\) and \((x\wedge y) \odot (x\wedge y)\) are in \(M\). It follows that \(s(x\wedge y)=1\) and therefore also
$$\begin{aligned} 1=s(x\wedge y)=s(x\wedge y)\odot s(x\wedge y)= s((x\wedge y) \odot (x\wedge y))\le s(x\odot y)=s(x\cdot y). \end{aligned}$$

Lemma 6

Let \((E;+,0,1)\) be a lattice effect algebra, \(S\) a non-empty set of E-semi-states on \({E}\). Then
  1. (a)

    the pointwise meet \(t=\bigwedge S\) is an E-semi-state on \({E}\),

     
  2. (b)

    if \(S\) is linearly ordered then \(q=\bigvee S\) is an E-semi-state on \({E}\).

     

Proof

(a): Let us check the conditions (i)–(iv) from Definition 4.

(i): Clearly, \(t(0)=\bigwedge \{s(0) \mid s\in S\} =\bigwedge \{0 \mid s\in S\}=0\) and \(t(1)=\bigwedge \{s(1) \mid s\in S\} =\bigwedge \{1 \mid s\in S\}=1\).

(ii): Assume that \(x\le y'\).There is an element \(s_0\in S\) such that \(s_0(x)+s_0(y)\) is defined and clearly \(t\le s_0\). It follows that \(t(x)+t(y)\) is defined. Let us compute the following
$$\begin{aligned} \begin{array}{r c l} t(x)+ t(y)&{}=&{}t(x)\oplus t(y)=\bigwedge \{s_1(x) \mid s_1\in S\} \oplus \bigwedge \{ s_2(y) \mid s_2\in S\}\\ &{}=&{}\bigwedge \{s_1(x) \oplus s_2(y) \mid s_1, s_2\in S\} \le \bigwedge \{s(x) \oplus s(y) \mid s\in S\}\\ &{}=&{}\bigwedge \{s(x) + s(y) \mid s\in S\}\le \bigwedge \{s(x+ y) \mid s\in S\}= t (x+ y). \end{array} \end{aligned}$$
(iii), (iv): Since \([0,1]\) is linearly ordered we have (by taking in the respective part of the proof either the minimum of \(s_1(x)\) and \(s_2(x)\) or the maximum of \(s_1(x)\) and \(s_2(x)\))
$$\begin{aligned} \begin{array}{r c l} t(x)\odot t(x)\!&{}=&{}\!\bigwedge \{s_1(x) \mid s_1\!\in \! S\} \!\odot \! \bigwedge \{ s_2(x) \mid s_2\!\in \! S\}\!=\!\bigwedge \{s_1(x) \odot s_2(x) \mid s_1, s_2\in S\} \\ &{}\ge &{}\bigwedge \{s(x) \odot s(x) \mid s\in S\}\!=\!\bigwedge \{s(x\odot x) \mid s\in S\}\!=\! t (x\odot x), \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{r c l} t(x)\odot t(x)\!&{}=&{}\!\bigwedge \{s_1(x) \mid s_1\!\in \! S\} \odot \bigwedge \{ s_2(x) \mid s_2\!\in \! S\}\!=\!\bigwedge \{s_1(x) \odot s_2(x) \mid s_1, s_2\in S\} \\ &{}\le &{}\bigwedge \{s(x) \odot s(x) \mid s\in S\}=\bigwedge \{s(x\odot x) \mid s\in S\}= t (x\odot x), \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{r c l} t(x)\oplus t(x)\!&{}=&{}\!\bigwedge \{s_1(x) \mid s_1\!\in \! S\} \oplus \bigwedge \{ s_2(x) \mid s_2\!\in \! S\}\!=\!\bigwedge \{s_1(x) \oplus s_2(x) \mid s_1, s_2\in S\} \\ &{}\ge &{}\bigwedge \{s(x) \oplus s(x) \mid s\in S\}=\bigwedge \{s(x\oplus x) \mid s\in S\}= t (x\oplus x), \end{array} \end{aligned}$$
and
$$\begin{aligned} \begin{array}{r c l} t(x)\oplus t(x)\!&{}=&{}\!\bigwedge \{s_1(x) \mid s_1\!\in \! S\} \oplus \bigwedge \{ s_2(x) \mid s_2\!\in \! S\}\!=\!\bigwedge \{s_1(x) \oplus s_2(x) \mid s_1, s_2\!\in \! S\} \\ &{}\le &{}\bigwedge \{s(x) \oplus s(x) \mid s\in S\}=\bigwedge \{s(x\oplus x) \mid s\in S\}= t (x\oplus x). \end{array} \end{aligned}$$
(b): As above we have to verify the conditions (i)–(iv).

(i): Clearly, \(q(0)=\bigvee \{s(0) \mid s\in S\} =\bigvee \{0 \mid s\in S\}=0\) and \(q(1)=\bigvee \{s(1) \mid s\in S\} =\bigvee \{1 \mid s\in S\}=1\).

(ii): Assume that \(x\le y'\).Then, for all \(u, v\in S\), \(u\le v\) we have \(u(x)\le v(x)\le v(y)'\). This yields that \(u(x)\le \bigwedge \{w(y)' \mid w\in S, u\le w\}= (\bigvee \{w(y) \mid w\in S, u\le w\})'=q(y)'\). It follows that \(q(x)\le q(y)'\), i.e., \(q(x)+q(y)\) is defined. Let us compute the following
$$\begin{aligned} \begin{array}{r c l} q(x)+ q(y)&{}=&{}\bigvee \{s_1(x) \mid s_1\in S\} + \bigvee \{ s_2(y) \mid s_2\in S\}\\ &{}=&{}\bigvee \{s_1(x) + s_2(y) \mid s_1, s_2\in S\} \le \bigvee \{s(x) + s(y) \mid s\in S\}\\ &{}\le &{}\bigvee \{s(x+ y) \mid s\in S\}= q (x+ y). \end{array} \end{aligned}$$
(iii), (iv): It follows by the same arguments as above in part (a) since \(\oplus \) and \(\odot \) distribute in any complete MV-algebra over arbitrary joins. \(\square \)

3.1 Dyadic Numbers and MV-Terms

We remark on some concepts introduced by Teheux in [23]. The set \(\mathbb D\) of dyadic numbers is the set of the rational numbers that can be written as a finite sum of integer powers of 2. We denote by \(f_0(x)\) and \(f_1(x)\) the terms \(x\oplus x\) and \(x\odot x\) respectively, and by \(T_{\mathbb D}\) the clone generated by \(f_0(x)\) and \(f_1(x)\).

Here, a clone is a set \(C\) of finitary operations on a set \(A\) such that
  • \(C\) contains all the projections \(\pi _{k}^{n}: A^n \rightarrow A\), defined by \(\pi _{k}^{n}(x_1, \ldots ,x_n) = x_k\),

  • \(C\) is closed under (finitary multiple) composition: if \(f, g_1, \ldots , g_m\) are members of \(C\) such that \(f\) is an \(m\)-ary operation, and \(g_j\) is an \(n\)-ary operation for every \(j\), then the \(n\)-ary operation \(h(x_1,\ldots ,x_n):= f(g_1(x_1, \ldots ,x_n), \ldots , g_m(x_1,\ldots ,x_n))\) is in \(C\).

We also denote by \(g_{.}\) the mapping between the set of finite sequences of elements of \(\{0, 1\}\) (and thus of dyadic numbers in \([0, 1]\)) and \(T_{\mathbb D}\) defined by:
$$\begin{aligned} g_{(a_1,\ldots ,a_k)} = f_{a_k} \circ \cdots \circ f_{a_1} \end{aligned}$$
for any finite sequence \((a_1, \ldots , a_k)\) of elements of \(\{0, 1\}\). If \(a = \sum ^{k}_{i=1} a_i 2^{-i}\), we sometimes write \(g_a\) instead of \(g_{(a_1,\ldots ,a_k)}\).

Lemma 7

[23, Lemma 1.14] If \(a^{*} = (a_i)_{i \in \mathbb N}\) and \(x^{*} = (x_i )_{i \in \mathbb N}\) are dyadic decompositions of two elements of \(a, x\in [0, 1]\), then, for any positive integer \({k\in \mathbb N}\),
$$\begin{aligned} g_{{\ulcorner {a^{*}}\urcorner }_{k}} (x) = \left\{ \begin{array}{l@{\quad }l} 1&{} \text {if}\ x > \sum ^{k}_{i=1} a_i 2^{- i} + 2^{-k}\\ 0&{} \text {if}\ x < \sum ^{k}_{i=1} a_i 2^{- i} \\ \sum ^{\infty }_{i=1} x_{i+k} 2^{-i}&{} \text {otherwise.} \end{array} \right. \end{aligned}$$
Let us define, for any \(m\in \mathbb {N}\) a term \(\mu _{m}(x)\in T_{\mathbb D}\). First, if \(m=1\) we put \(\mu _{1}(x)=f_1(x)\). Second, assume that \(\mu _{m}(x)\) is defined. Then we put \(\mu _{m+1}(x)=\mu _1(\mu _{m}(x))\). It follows that, evaluating \(\mu _{m}(x)\) for \(x=\sum ^{\infty }_{i=1} x_{i} 2^{-i}\) in the standard MV-algebra \([0,1]\), we obtain by Lemma 7
$$\begin{aligned} \mu _{m}(x)=g_{\underbrace{(0, \ldots , 0)}_{m\mathrm{\small -times}}} (x) = \left\{ \begin{array}{l@{\quad }l} 1&{} \text {if}\ x > 2^{-m}\\ \sum ^{\infty }_{i=1} x_{i+m} 2^{-i}&{} \text {otherwise.} \end{array} \right. \end{aligned}$$
We will also need the following:

Corollary 8

[23, Corollary 1.15 (1)] Let us have the standard MV-algebra \([0,1]\), \(x\in [0, 1]\) and \(r\in (0,1)\cap {\mathbb D}\). Then there is a term \(t_r\) in \(T_{\mathbb D}\) such that
$$\begin{aligned} t_r(x)=1 \quad \text {if and only if}\quad r\le x. \end{aligned}$$

Proposition 9

[2, Proposition 1] Let \({A}\) be a linearly ordered MV-algebra, \(s: A\rightarrow [0,1]\) an MV-morphism, \(x\in A\). Then \(s(x)=1\) iff \(t_r(x)=1\) for all \(r\in (0,1)\cap {\mathbb D}\).

Equivalently, \(s(x)<1\) iff there is a dyadic number \(r\in (0,1)\cap {\mathbb D}\) such that \(t_r(x)\not =1\). In this case, \(s(x)<r\).

Following the preceding corollary we are able to compare E-semi-states \(s\) via inclusion relation looking only on the respective order filters \(s^{-1}(\{1\}\).

Lemma 10

Let \((E;+,0,1)\) be a lattice effect algebra, \(s, t\) E-semi-states on \({E}\). Then \(t\le s\) iff \(t(x)=1\) implies \(s(x)=1\) for all \(x\in E\).

Proof

Clearly, \(t\le s\) yields the condition \(t(x)=1\) implies \(s(x)=1\) for all \(x\in E\).

Assume now that \(t(x)=1\) implies \(s(x)=1\) for all \(x\in E\) is valid and that there is \(y\in E\) such that \(s(y) < t(y)\). Thus, there is a dyadic number \(r\in (0,1)\cap \mathbb D\) such that \(s(y)<r< t(y)\). By Corollary 8 there is a term \(t_r\) in \(T_{\mathbb D}\) such that \(t_r(s(y))< 1\) and \(t_r(t(y))=1\). It follows that \(s(t_r(y))=t_r(s(y))< 1\) and \(t(t_r(y))=t_r(t(y))=1\). The last condition yields that \(s(t_r(y))=1\), a contradiction.

3.2 Riesz Decomposition Property and Ideals in Effect Algebras

We recall that an effect algebra \({E}\) satisfies the Riesz Decomposition Property ((RDP) in abbreviation) if \(x \le y_1 + y_2\) implies that there exist two elements \(x_1, x_2 \in E\) with \(x_1 \le y_1 \) and \(x_2 \le y_2\) such that \(x = x_1 + x_2\). Any MV-algebra satisfies (RDP).

An ideal of an effect algebra \({E}\) is a non-empty subset \(I\) of \(E\) such that
  1. (i)

    \(x \in E\), \(y \in I\), \(x\le y\) imply \(x \in I\),

     
  2. (ii)

    if \(x,y \in I\) and \(x+y\) is defined in \({E}\), then \(x+y \in I\).

     
A filter of an effect algebra \({E}\) is an ideal in the dual effect algebra \({E}^{op}=(E;\cdot ,1,0)\).

We denote by \({Id}({E})\) the set of all ideals of \({E}\). An ideal \(I\) is said to be a Riesz ideal if, for \(x \in I\), \(a,b \in E\) and \(x \le a+b\), there exist \(a_1,b_1 \in I\) such that \(x = a_1 +b_1\) and \(a_1 \le a\) and \(b_1 \le b\).

For example, if \({E}\) has (RDP), then any ideal of \({E}\) is Riesz and there is a one-to-one correspondence between congruences and ideals of \(E\) (see [13]), and given an ideal \(I\), the congruence \(\sim _I\) on \({E}\) is assigned by \(a\sim _I b\) iff there are \(x,y \in I\) with \(x\le a\) and \(y\le b\) such that \(a-x = b-y\). Then the quotient \({E}/I\) is an effect algebra with RDP.

The main supporting statement that we need (and which is essentially contained in the proof of [17, Theorem 4.5] or in [2, Proposition 3]) is

Proposition 11

Let \((E;+,0,1)\) be a lattice effect algebra with an order reflecting set \(S=\{ s: E\rightarrow [0,1]\mid \ s\ \text {is an}\ \text {E-state on}\)\({E}\}\), \(t\) a Jauch–Piron E-semi-state on \({E}\) and \(S_{t}=\{ s\in S \mid s\ge t\}\). Then \(t=\bigwedge S_t\).

Proof

We may assume that \(E\subseteq [0,1]^{S}\) such that \(x+y\), \(x\cdot y\), \(x\oplus x\) and \(x\odot x\) computed in \({E}\) gives us the same results as \(x+y\), \(x\cdot y\), \(x\oplus x\) and \(x\odot x\) computed in \([0, 1]^{S}\) and restricted to elements from \(E\). This means that the inclusion map \(i:{E}\rightarrow [0, 1]^{S}\) is an order reflecting E-morphism of lattice effect algebras. Note also that \([0, 1]^{S}\) is a lattice effect algebra with RDP (MV-algebra).

Clearly, \(t\le \bigwedge S_t\). Assume that there is \(x\in E\) such that \(t(x)<\bigwedge S_t (x)\). Thus, there is a dyadic number \(r\in (0,1)\cap \mathbb D\) such that \(t(x)<r<\bigwedge S_t (x).\) Again by Corollary 8 there is a term \(t_r\) in \(T_{\mathbb D}\) such that \(t(t_r(x))=t_r(t(x))<1\). Let us put \(U=\{z\in E\mid t(z)=1\}\). The set \(U\) is by Lemma 5 a filter of \({E}\) which is closed under finite meets and \(z\in U\) yields \(\mu _k(z)\in U\) for all \(k\in \mathbb {N}\) since \(t\) is a Jauch–Piron E-semi-state, \(t_r(x)\not \in U\). Let \(V\) be a filter of \([0, 1]^{S}\) generated by the set \(U\). Then by [13, Proposition 3.1]
$$\begin{aligned} \begin{array}{r l} V=\{y\in [0, 1]^{S} \mid &{}\exists n\in \mathbb {N}, \exists g_1, \ldots , g_n\in [0, 1]^{S}, \exists f_1, \ldots , f_n\in U,\\ &{}f_i\le g_i, i=1, \ldots , n, y=g_1\cdot {} \ldots {}\cdot g_n\}. \end{array} \end{aligned}$$
Let us assume that \(t_r(x)\in V\). Then \(\exists n\in \mathbb {N}, \exists g_1, \ldots , g_n\in [0, 1]^{S}, \exists f_1, \ldots , f_n\in U, f_i\le g_i, i=1, \ldots , n, t_r(x)=g_1\cdot {} \ldots {}\cdot g_n\). We then put \(f=f_1\wedge \dots \wedge f_n\). Let \(k\in \mathbb {N}\) be minimal such that \(n\le 2^{k}\). It follows that \(t_r(x)=g_1\cdot {} \dots {}\cdot g_n \cdot \underbrace{1 \cdot {} \dots {}\cdot 1}_{2^{k}-n\ \text {times}}\ge \mu _{k}(f)\in U\), a contradiction.

So we have that \(t_r(x)\notin V\). Let \(W\) be a maximal filter of \([0, 1]^{S}\) which does contain \(V\) and does not contain \(t_r(x)\). Then the set \(I=\{y\in [0, 1]^{S} \mid y'\in W\}\) is a prime ideal in \([0, 1]^{S}\), \(t_r(x)'\notin I\). It follows by [13, Proposition 6.5 and Proposition 6.10] that \([0, 1]^{S}/I\) is a linearly ordered effect algebra, i.e. an MV-algebra such that \(t_r([x]_{I})=[t_r(x)]_{I}\not =[1]_I\). In particular the factor map \(\pi _I:[0, 1]^{S}\rightarrow [0, 1]^{S}/I\) is an MV-morphism such that \(\pi _I(x)=[x]_{I}\) and \(\pi _I(U)\subseteq \pi _I(V)\subseteq \pi _I(W)=\{[1]_I\}\).

Let us denote by \(U_I\) the maximal ideal of \( [0, 1]^{S}/I\) and by \(\overline{s}: [0, 1]^{S}/I \rightarrow [0,1]\) the corresponding MV-morphism. Hence, by Proposition 9, \(\overline{s}([x]_{I})<r<1\). Let us put \(s=\overline{s}\circ \pi _I\circ i\). Then \(s\) is an E-state such that \(s(U)=\overline{s}(\pi _I(U))=\overline{s}(\{[1]_I\})=\{1\}\). It follows by Lemma 10 that \(t\le s\), i.e., \(s\in S_t\) and \(s(x)=\overline{s}([x]_{I})< r <\bigwedge S_t (x) \le s(x)\), a contradiction.

4 Functions Between Lattice Effect Algebras and Their Construction

This section studies the notion of an EM-function between lattice effect algebras and a very Jauch–Piron EM-function between lattice effect algebras. The overall goal of this section is to establish in some sense a canonical construction of very Jauch–Piron EM-functions. We would like to understand this construction because its particular case is the construction of S-tense operators when we apply the time frame \((T,R)\). This construction can serve as an ultimate source of numerous examples.

Definition 12

  1. (1)
    By an EM-function G between lattice effect algebras is meant a function \(G:{E}_1\rightarrow {E}_2\) such that \((E_1;+_1, 0_1,1_1)\) and \(({E}_2;+_2, 0_2,1_2)\) are lattice effect algebras and
    • (EM1) \(G(0_1)=0_2\), \(G(1_1)=1_2\),

    • (EM2) \(G(x)+_2 G(y)\le G(x+_1 y)\),

    • (EM3) \(G(x)\odot _2 G(x)=G(x\odot _1 x)\),

    • (EM4) \(G(x)\oplus _2 G(x)= G(x\oplus _1 x)\).

    If \({E}_1={E}_2\) we say that \(G\) is an EM-operator on \({E}_1\).
     
  2. (2)
    If moreover \(G\) satisfies conditions
    • (EM2)’ \(G(x)+_2 G(y)= G(x+_1 y)\)

    we say that \(G\) is an EM-morphism between lattice effect algebras.
     
  3. (3)
    If \(G\) is an EM-function between lattice effect algebras such that (EM5) (resp. (EM5)’) is satisfied we say that \(G\) is Jauch–Piron (resp. very Jauch–Piron)
    • (EM5) \(G(x)=1_2=G(y)\) implies \(G(x\wedge _1 y)=1_2\),

    • (EM5)’ \(G(x)\wedge _2 G(y)= G(x\wedge _1 y)\).

     
  4. (4)
    If \(G\) is an EM-function between lattice effect algebras such that
    • (EM6) \(G(x)\odot _2 G(y)\le G(x\odot _1 y)\),

    • (EM7) \(G(x)\oplus _2 G(y)\le G(x\oplus _1 y)\),

    • (EM8) \(G(x^{n})=G(x)^{n}\) for all \(n\in {\mathbb N}\),

    • (EM9) \(n\times _2 G(x)= G(n\times _1 x)\) for all \(n\in {\mathbb N}\),

    we say that \(G\) is an FEM-function.
     
  5. (5)

    If \(G:{E}_1\rightarrow {E}_2\) and \(H:{B}_1\rightarrow {B}_2\) are EM-functions between lattice effect algebras, then a morphism between G and H is a pair \((\varphi ,\psi )\) of morphisms of lattice algebras \(\varphi :{E}_1 \rightarrow {B}_1\) and \(\psi :{E}_2 \rightarrow {B}_2\) such that \(\psi (G(x))=H(\varphi (x))\), for any \(x\in E_1\).

     
Note that (EM2)’ yields (EM2), (EM5)’ yields (EM5). By essentially same considerations as in Lemma 5 the condition (EM5) yields the following:
$$\begin{aligned} G(x)=1_2=G(y)\hbox { and } x\cdot y \hbox { defined implies } G(x\odot _1 y)=1_2. \end{aligned}$$
Also, a composition of EM-functions (very Jauch–Piron EM-functions, FEM-functions) is an EM-function (a very Jauch–Piron EM-function, an FEM-function) again and a composition of a very Jauch–Piron EM-function with a Jauch–Piron EM-function is a Jauch–Piron EM-function.

The notion of an EM-function generalizes both the notions of an E-semi-state from Sect. 3 of this paper, of a \(\odot \)-operator from [17] for MV-algebras and of an fm-function from [2].

According to both (EM3) and (EM4), \(G|_{S({E}_1)}: {S({E}_1)} \rightarrow {S({E}_2)}\) is an EM-function (a Jauch–Piron EM-function, a very Jauch–Piron EM-function) whenever \(G\) has the corresponding property.

Recall also, that EM-functions between E-representable MV-algebras (which are exactly semisimple MV-algebras) coincide with FEM-functions between them (see [2, Proposition 6]). On the contrary, there is an MV-algebra \(M\) with an EM-operator on \(M\) that is not an FEM-operator (see [2, Proposition 7]).

Lemma 13

Let \(G:{E}_1\rightarrow {E}_2\) be an EM-function between lattice effect algebras, \(r\in (0,1)\cap {\mathbb D}\). Then \(t_r(G(x))= G(t_r(x))\) for all \(x\in E_1\).

Proof

Note that \(G(x)\oplus _2 G(x)= G(x\oplus _1 x)\) by (EM4) and \(G(x)\odot _2 G(x)= G(x\odot _1 x)\) by (EM3). Then, since \(t_r\in T_{\mathbb D}\) is defined inductively using only the operations \((-)\oplus (-)\) and \((-)\odot (-)\), we get \(t_r(G(x))=G(t_r(x))\).

By a frame is meant a triple \((S,T,R)\) where \(S,T\) are non-void sets and \(R\subseteq S\times T\). If \(S=T\), we will write briefly \((T, R)\) for the frame \((T,T,R)\). For our sake, we will assume that for every \(x\in S\) there is \(y\in T\) such that \(xRy\) and for every \(y\in T\) there is \(x\in S\) such that \(xRy\). Having a lattice algebra \((E;+,0,1)\) and a non-void set \(T\), we can produce the direct power \((E^T;+, o,j)\) where the operation \(+\) and the induced operations \(\vee \), \(\wedge \), \(\oplus \), \(\odot \) and \(\lnot \) are defined and evaluated on \(p,q\in E^T\) componentwise. Moreover, \(o, j\) are such elements of \(E^T\) that \(o(t)=0\) and \(j(t)=1\) for all \(t\in T\). The direct power \({E}^T\) is again a lattice effect algebra.

The notion of frame allows us to construct examples of FEM-functions between lattice effect algebras.

In what follows we will need the following

Lemma 14

([6, 7]) Let \((E;+,0,1)\) be an effect algebra. Let \(a_i, b_i, c_i\in E\) for \(i\in I\) and assume \(a_i\bot b_i\) for all \(i\in I\). Let \(\bigwedge \{a_i; i\in I\}\), \(\bigwedge \{b_i; i\in I\}\), \(\bigwedge \{c_i; i\in I\}\), \(\bigwedge \{c_i'; i\in I\}\) and \(\bigwedge \{a_i+b_i; i\in I\}\) exist. Then
  1. (1)
    \(\bigwedge \{a_i; i\in I\} + \bigwedge \{b_i; i\in I\}\) exists and
    $$\begin{aligned} \bigwedge \{a_i; i\in I\} + \bigwedge \{b_i; i\in I\}\le \bigwedge \{a_i+b_i; i\in I\} \end{aligned}$$
     
  2. (2)

    \(\bigwedge \{c_i'; i\in I\}\le (\bigwedge \{c_i; i\in I\})'\).

     

We now prove the generalization of [2, Theorem 2].

Theorem 15

Let \({M}\) be a linearly ordered complete MV-algebra, \((S, T, R)\) be a frame and \(G\) be a map from \({M}^T\) into \({M}^S\) defined by
$$\begin{aligned} G(p)(s) = \bigwedge \{p(t) \mid t \in T, sRt\}, \end{aligned}$$
for all \(p\in M^T\) and \(s\in S\). Then \(G\) is a very Jauch–Piron FEM-function between MV-algebras which has a left adjoint \(P\), i.e., \(P(q)\le p\) iff \(q\le G(p)\) for all \(q\in M^S\) and \(p\in M^T\). In this case, for all \(q\in M^S\) and \(t\in T\),
$$\begin{aligned} P(q)(t) = \bigvee \{q(s) \mid s \in T, sRt\} \end{aligned}$$
and \(P:({M}^S)^{op}\rightarrow ({M}^T)^{op}\) is a very Jauch–Piron FEM-function between MV-algebras.

Proof

Trivially we can verify \(G(o)(s)=0, G(j)(s)=1\) for all \(s\in S\) due to the fact that \(o(t)=0\) and \(j(t)=1\) for each \(t\in T\) thus (EM1) holds. Let us check (EM2). Assume that \(p,q\in M^T\) and \(p+q\) exists. Hence, \(p(t)+q(t)\) exists for each \(t\in T\). Let \(s\in S\). By Lemma 14 (1) also \(\bigwedge \{p(t); s R t\} + \bigwedge \{q(t); s R t\}\) exists and \(G(p)(s)+G(q)(s)=\bigwedge \{p(t); s R t\} + \bigwedge \{q(t); s R t\}\le \bigwedge \{p(t)+q(t); s R t\}=G(p+q)(s)\). Thus \(G(p)+G(q)\le G(p+q)\).

The conditions (EM5)’, (EM6)-(EM9) and the adjointness between \(P\) and \(G\) follow directly from [2, Theorem 2], the condition (EM3) follows from (EM8) and the condition (EM4) follows from (EM9). The remaining part for \(P\) follows by the same arguments applied to the dual MV-algebra \({M}^{op}\).

We say that \(G:{M}^T \rightarrow {M}^S\) is the canonical very Jauch–Piron FEM-function induced by the frame\((S,T,R)\)and the MV-algebra\({M}\).

Corollary 16

Let \({M}\) be a linearly ordered complete MV-algebra, \((S, R)\) be a frame, \(G\) and \(H\) be maps from \({M}^S\) into \({M}^S\) defined by
$$\begin{aligned} \begin{array}{l c l} G(p)(s) &{}=&{} \bigwedge \{p(t) \mid t \in S, sRt\},\\ H(p)(s) &{}=&{} \bigwedge \{p(t) \mid t \in S, tRs\} \end{array} \end{aligned}$$
for all \(p\in M^S\) and \(s\in S\). Then \(G\) (\(H\)) is a very Jauch–Piron E-tense operator on \({M}^S\) which has a left adjoint \(P\) (\(F\)) and \(({M}^S; G, H)\) is a tense effect algebra. In this case, for all \(q\in M^S\) and \(t\in S\),
$$\begin{aligned} \begin{array}{l c l} P(q)(t)&{} =&{} \bigvee \{q(s) \mid s \in S, sRt\},\\ F(q)(t)&{} =&{} \bigvee \{q(s) \mid s \in S, tRs\}. \end{array} \end{aligned}$$

Now we give the postponed proof of Theorem 1. It is enough to check conditions (T7) and (T8), the remaining parts follow immediately from Corollary 16. But (T7) and (T8) follow from [2, Theorem 6].

5 The Representation Theorem and its Applications

The aim of this section is to show that an E-representable E-Jauch–Piron lattice effect algebra with E-tense operators \(G\) and \(H\) (or with a very Jauch–Piron operator \(G\)) can be represented in a power of the standard MV-algebra \([0,1]\), where the set of all Jauch–Piron E-states serves as a time frame (in the sense given in Introduction) with a relation defined by means of the point-wise ordering of these states on \(x\) as well as on \(G(x)\) or \(H(x)\). It properly means that for every E-representable E-Jauch–Piron lattice effect algebra with tense operators a suitable time frame exists and can be constructed by use of the previously introduced concepts.

Theorem 17

Let \(G: E_1\rightarrow E_2\) be a very Jauch–Piron EM-function between lattice effect algebras. Let \( E_1\) be an E-representable lattice effect algebra and let \( E_2\) be an E-Jauch–Piron representable lattice effect algebra, \(T\) a set of all E-states from \( E_1\) to the standard MV-algebra \([0,1]\) and \(S\) a set of all Jauch–Piron E-states from \( E_2\) to \([0,1]\).

Further, let \((S, T, \rho _G)\) be a frame such that the relation \(\rho _G\subseteq S\times T\) is defined by
$$\begin{aligned} s\rho _G t \text{ if } \text{ and } \text{ only } \text{ if } s(G(x))\le t(x) \text{ for } \text{ any } x\in E_1. \end{aligned}$$
Then \(G\) is representable via the canonical very Jauch–Piron FEM-function \(G^*:[0,1]^T \rightarrow [0,1]^S\) between MV-algebras induced by the frame \((S,T,\rho _G)\) and the standard MV-algebra \([0,1]\), i.e., the following diagram of EM-functions commutes:

Proof

Assume that \(x\in E_1\) and \(s\in S\). Then \({i_{ E_2}^S}(G(x))(s)=s(G(x))\le t(x)\) for all \(t\in T\) such that \((s,t)\in \rho _G\). It follows that \({i_{ E_2}^S}(G(x))\le G^{*}({i_{ E_1}^T}(x))\).

Note that \(s\circ G\) is a Jauch–Piron semi-state on \( E_1\) and by Proposition 11 we get that
$$\begin{aligned} \begin{array}{r c l} s\circ G&{}=&{}\bigwedge \{ t: {E}_1\rightarrow [0,1]\mid \ t\ \text {is an}\ \text {E-state}, t\ge s\circ G\}\\ &{}=&{}\bigwedge \{ t\in T\mid \ (s, t)\in \rho _G\}. \end{array} \end{aligned}$$
This yields that actually \({i_{ E_2}^S}(G(x))= G^{*}({i_{ E_1}^T}(x))\).

Note also that [2, Theorem 3] is a particular case of Theorem 15.

The following result which is an immediate corollary of Theorem 17 generalizes the main result of the paper [17, Theorem 4.5].

Corollary 18

(Representation theorem for lattice effect algebras with a very Jauch–Piron operator) For any E-representable E-Jauch–Piron lattice effect algebra \( E\) with a very Jauch–Piron operator \(G\) and where T is an order reflecting set of all Jauch–Piron E-states, \( E\) is embeddable via E-morphism \(i_{ E}^{T}\) into the canonical MV-algebra \({ L}_G=([0,1]^{T};G^{*})\) with a very Jauch–Piron operator \(G^{*}\) induced by the canonical frame \((T,\rho _G)\) and the standard MV-algebra \([0,1]\). Further, for all \(x\in E\) and for all \(s\in T\), \(s(G(x))=G^{*}((t(x))_{t\in T})(s)\).

The next Theorem yields a solution of the representation problem for E-tense operators. The idea of the proof is taken from [2, Theorem 5].

Theorem 19

Let \( E\) be an E-representable E-Jauch–Piron lattice effect algebra with E-tense operators \(G\) and \(H\). Then \(( E,G,H)\) can be embedded into the tense MV-algebra \(([0,1]^T,G^*,H^*)\) induced by the frame \((T,\rho _G),\) where \(T\) is the set of all Jauch–Piron E-states from \( E\) to \([0,1]\) and the relation \(\rho _G\) is defined by
$$\begin{aligned} s\rho _G t \text{ if } \text{ and } \text{ only } \text{ if } s(G(x))\le t(x) \text{ for } \text{ any } x\in E. \end{aligned}$$

Proof

We may assume that \(E\subseteq [0,1]^T\). First, let us define a second relation \(\rho _H\subseteq T^2\) by the stipulation:
$$\begin{aligned} t\rho _H s \text{ if } \text{ and } \text{ only } \text{ if } t(H(x))\le s(x) \text{ for } \text{ any } x\in E. \end{aligned}$$
We show that the equality \(\rho _G=\rho _H^{-1}\) holds. Let us suppose that \(s\rho _G t\) for some \(s,t\in T.\) Due to the definition of E-tense operators we have \(G(H(x)')'\le x\) and hence \(x'\le G(H(x)')\), i.e., \(s(x')\le s(G(H(x)'))\). Then \(s\rho _G t\) yields \(s(G(H(x)'))\le t(H(x)')\) and together with the preceding we get \(s(x')\le t(H(x)')\). It follows that \(t(H(x))\le s(x)\) for any \(x\in E.\)

Due to the definition of \(\rho _H\) we have \(t\rho _H s\) and \(\rho _G\subseteq \rho _H^{-1}.\) Analogously we can prove the second inclusion.

The remaining part follows from Theorem 17. Basically, the obtained equations \(G^*(x) = G(x)\) and \(H^*(x)=H(x)\) finish the proof.

Now we give the postponed proof of Theorem 3. It is enough to show that the relation \(\rho _G\) from Theorem 19 is reflexive and transitive. But to prove this we can use the same arguments as in [2, Theorem 6] or [16, Lemma 4.7].

The next example shows an E-representable E-Jauch–Piron lattice effect algebra \(E\), that is not an MV-algebra, with all possible (E- and S-)tense operators. In particular, we constructed a (time) frame \((T,\rho )\) from the set \(T\) of all Jauch–Piron E-states on \(E\) and we also described a respective representation in \([0,1]^{T}\), in accordance to Theorem 19.

Example 20

Let us consider a lattice effect algebra \((E;+,0,1)\), where \(E = \{0, a, b, c, a + b = 2c = 1\}\) (see Fig. 1).

It is a horizontal sum of a Boolean algebra \(\{0, a, b, a + b = 1\}\) and an MV-chain \(\{0, c, 2c = 1\}\). Let us have pairs of tense operators \(G_{i}, H_{i}\) on \({ E}\) for some \(i \in I \subseteq \mathbb {N}\). By definition we have \(G_i(1) = H_i(1)=1\) and \(G_{i}(0)=H_i(0)=0\) for any \(i \in I\). For every \(i \in I\), it holds \(G_i(c) \in \{0,c\}\). Moreover, if \(G_{i}(c) = c\) then \(H_{i}(c) = c\) and \(G_{i}(c) = 0\) implies \(H_{i}(c) = 0\) (see Fig. 2 for i = 2).
  • \(G_{1} = H_{1} = \mathrm{id}_{E}\),

  • \(G_{2}(a) = a, G_{2}(b) =0, G_{2}(c) = c \), \(H_{2}(a) = 0, H_{2}(b) =b, H_{2}(c) = c\),

  • \(G_{3}(a) = 0, G_{3}(b) =b, G_{3}(c) = c\), \(H_{3}(a) = a, H_{3}(b) =0, H_{3}(c) = c\),

  • \(G_{4}(a) = 0, G_{7}(b) =0, G_{7}(c) = c\), \(H_{7}(a) = 0, H_{7}(b) =0, H_{7}(c) = c\)

  • \(G_{5}(a) = b, G_{4}(b) =a, G_{4}(c) = c\), \(H_{4}(a) = b, H_{4}(b) =a, H_{4}(c) = c\),

  • \(G_{6}(a) = b, G_{5}(b) =0, G_{5}(c) = c\), \(H_{5}(a) = b, H_{5}(b) =0, H_{5}(c) = c\),

  • \(G_{7}(a) = 0, G_{6}(b) =a, G_{6}(c) = c\), \(H_{6}(a) = 0, H_{6}(b) =a, H_{6}(c) = c\),

Additional pairs of tense operators \(G_{j+7}, H_{j+7}\) can be defined by setting \(G_{j+7}(c) = H_{j+7}(c) = 0\) and \(G_{j+7}(a)= G_{j}(a), H_{j+7}(a)= H_{j}(a), G_{j+7}(b)= G_{j}(b), H_{j+7}(b)= H_{j}(b)\) for any \(j \in \{1, \ldots , 7\}\). From equalities \(a \oplus a = a\), \(b \oplus b = b\) and \(c \oplus c = c+c=1\) it follows, that the properties (T5) and (T6) are satisfied if and only if \(G_{i}(c) = H_{i}(c) = c\), hence E-tense operators are only \(G_1, \ldots G_7, H_1 \ldots H_7\). The axioms (T7) and (T8), which are necessary for E-tense operators to be S-tense, hold only for \(G_1, \ldots G_4, H_1 \ldots H_4\).

There exist two extremal states (i.e., they cannot be written as a non trivial convex combination of other states) \(T = \{s_{1}, s_{2}\}\), given by \(s_1(a) = 1, s_{1}(b) = 0\), \(s_2(a)=0, s_2(b) =1\) and \(s_1(c) = s_2(c) = \frac{1}{2}\). The set \(T\) is also the set of all E-states, it is order reflecting and \(s_1, s_2\) are Jauch–Piron, hence \({ E}\) is an E-representable E-Jauch–Piron lattice effect algebra.

By Theorem 19 for every E-representable E-Jauch–Piron lattice effect algebra \(({ E}, G_{i}, H_{i})\) with E-tense operators there exists an embedding into a tense MV-algebra \(([0,1]^T, G_{i}^{*}, H_{i}^{*})\) with a frame \((T, \rho _{G_{i}})\), where relation \(\rho _{G_{i}}\) is given by the following:
  • \(\rho _{G_{1}}:\)\(s_1 \rho _{G_{1}} s_1\), \(s_2 \rho _{G_{1}} s_2\),

  • \(\rho _{G_{2}}:\)\(s_1 \rho _{G_{2}} s_1\), \(s_2 \rho _{G_{2}} s_1\), \(s_2 \rho _{G_{2}} s_2\),

  • \(\rho _{G_{3}}:\)\(s_1 \rho _{G_{3}} s_1\), \(s_1 \rho _{G_{3}} s_2\), \(s_2 \rho _{G_{3}} s_2\),

  • \(\rho _{G_{4}}:\)\(s_1 \rho _{G_{7}} s_1\), \(s_2 \rho _{G_{4}} s_1\), \(s_2 \rho _{G_{7}} s_1\), \(s_2 \rho _{G_{7}} s_2\).

  • \(\rho _{G_{5}}:\)\(s_1 \rho _{G_{4}} s_2\), \(s_2 \rho _{G_{5}} s_1\),

  • \(\rho _{G_{6}}:\)\(s_1 \rho _{G_{5}} s_1\), \(s_1 \rho _{G_{6}} s_2\), \(s_2 \rho _{G_{5}} s_1\),

  • \(\rho _{G_{7}}:\)\(s_2 \rho _{G_{6}} s_2\), \(s_1 \rho _{G_{7}} s_2\), \(s_2 \rho _{G_{6}} s_1\),

We can see, that relation \(\rho _i\) is reflexive and transitive iff \(i \in \{ 1, \ldots , 4\}\), that is for the case when \(G_i, H_i\) are S-tense operators. The embedding of \({ E}\) into \([0,1]^{T}\) is then given by \(i_{ E}^{T}(a) = (1,0)\), \(i_{ E}^{T}(b) = (0,1)\) and \(i_{ E}^{T}(c) = (\frac{1}{2},\frac{1}{2})\). As an example, let us investigate \(G_{3}^{*}\) and \(H_{3}^{*}\).
  • \(G_{3}^{*}(i_{ E}^{T}(a)) = G_{3}^{*}(1,0) = (\bigwedge \{t(a) \mid t \in \{s_1,s_2\}, s_1 \rho _{3} t\}, \bigwedge \{t(a) \mid t \in \{s_1,s_2\}, s_2 \rho _{3} t\}) = (s_2(a),s_2(a)) = (0,0)\),

  • \(G_{3}^{*}(i_{ E}^{T}(b)) = G_{3}^{*}(0,1) = (\bigwedge \{t(b) \mid t \in \{s_1,s_2\}, s_1 \rho _{3} t\}, \bigwedge \{t(b) \mid t \in \{s_1,s_2\}, s_2 \rho _{3} t\}) = (s_1(b),s_2(b)) = (0,1)\),

  • \(G_{3}^{*}(i_{ E}^{T}(c)) = G_{3}^{*}(\frac{1}{2},\frac{1}{2}) = (\frac{1}{2},\frac{1}{2})\),

  • \(H_{3}^{*}(i_{ E}^{T}(a)) = H_{3}^{*}(1,0) = (\bigwedge \{t(a) \mid t \in \{s_1,s_2\}, t \rho _{3} s_1\}, \bigwedge \{t(a) \mid t \in \{s_1,s_2\}, t \rho _{3} s_2\}) = (s_1(a),s_2(a)) = (1,0)\),

  • \(H_{3}^{*}(i_{ E}^{T}(b)) = H_{3}^{*}(0,1) = (\bigwedge \{t(b) \mid t \in \{s_1,s_2\}, t \rho _{3} s_1\}, \bigwedge \{t(b) \mid t \in \{s_1,s_2\}, t \rho _{3} s_2\}) = (s_1(b),s_1(b)) = (0,0)\),

  • \(H_{3}^{*}(i_{ E}^{T}(c)) = H_{3}^{*}(\frac{1}{2},\frac{1}{2}) = (\frac{1}{2},\frac{1}{2})\).

Fig. 1

\((E;+,0,1)\)

Fig. 2

Example of S-tense operators \(G_3, H_3\)

There can be defined additional nine relations on the set \(T\), namely \({\mathcal R} = \{\emptyset \), \(\{(s_1,s_1)\}\), \(\{(s_1,s_2) \}\), \(\{(s_2,s_1) \}\), \(\{(s_2,s_2) \}\), \(\{(s_1,s_1),(s_1,s_2) \}\), \(\{(s_2,s_2),(s_2,s_1)\}\), \(\{(s_1,s_1),(s_2,s_1)\}\), \(\{(s_2,s_2),(s_1,s_2) \}\}\). For any relation \(\delta \in {\mathcal R}\), the maps \(G^{*}_{\delta }, H^{*}_{\delta }\) which are defined as in Corollary 16 by the prescriptions
$$\begin{aligned} \begin{array}{l c l} G^{*}_{\delta }(p)(s) &{}=&{} \bigwedge \{p(t) \mid t \in T, s\delta t\},\\ H^{*}_{\delta }(p)(s) &{}=&{} \bigwedge \{p(t) \mid t \in T, t\delta s\} \end{array} \end{aligned}$$
for all \(p\in [0,1]^T\) and \(s\in T\) will not satisfy the axiom (T1). Hence \(G^{*}_{\delta }, H^{*}_{\delta }\) are not a pair of tense operators on \([0,1]^T\). Moreover, for any \(\sigma \in {\mathcal R}, \sigma \not = \emptyset \), it holds \(G^{*}_{\sigma }(i_{ E}^{T}(c)) = G^{*}_{\sigma }(\frac{1}{2},\frac{1}{2}) \notin i_{ E}^{T}(E)\) or \(H^{*}_{\sigma }(\frac{1}{2},\frac{1}{2}) \notin i_{ E}^{T}(E)\), \(i_{ E}^{T}(E) \subseteq [0,1]^T\), i.e., \(G^{*}_{\sigma }, H^{*}_{\sigma }\) define only partial (but not partial tense) operators \(G_{\sigma }, H_{\sigma }\) on \(E\).

6 Conclusion

Note that any semisimple MV-algebra is an E-representable E-Jauch–Piron lattice effect algebra. It follows that our result generalizes the results obtained in [2, 17]. Also, if we introduce modal operators on lattice effect algebras in accordance with [1] or [5] our method yields their representation on E-representable E-Jauch–Piron lattice effect algebras.

The basic results are
  1. (i)

    a description of Jauch–Piron E-semi-states on lattice effect algebras with an order reflecting set \(T\) of E-states as meets of these E-states.

     
  2. (ii)

    A representation theorem for E-representable E-Jauch–Piron lattice effect algebras with E-tense operators.

     
In addition, we have presented an illustrating example of our method.

Notes

Acknowledgments

We thank the anonymous referees for the careful reading of the paper and the suggestions on improving its presentation. All authors acknowledge the support by ESF Project CZ.1.07/2.3.00/20.0051 Algebraic methods in Quantum Logic of the Masaryk University.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Algebra and Geometry, Faculty of SciencePalacký University OlomoucOlomoucCzech Republic
  2. 2.Department of Mathematics and Statistics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic

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