# Left residuated operators induced by posets with a unary operation

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## Abstract

The concept of operator left residuation has been introduced by the authors in their previous paper (Chajda and Länger in Asian Eur J Math 11:1850097, 2018). Modifications of so-called quantum structures, in particular orthomodular posets, like pseudo-orthomodular, pseudo-Boolean and Boolean posets are investigated here in order to show that they are operator left residuated or even operator residuated. In fact, they satisfy more general sufficient conditions for operator residuation assumed for bounded posets equipped with a unary operation. It is shown that these conditions may be also necessary if a generalized version using subsets instead of single elements is considered. The above-listed posets can serve as an algebraic semantics for the logic of quantum mechanics in a broad sense. Moreover, our approach shows connections to substructural logics via the considered residuation.

## Keywords

Operator residuation Operator left adjointness Boolean poset Pseudo-Boolean poset Pseudo-orthomodular poset Generalized operator residuation## 1 Introduction

It was shown by Birkhoff and von Neumann (1936) and, independently, by Husimi (1937) that orthomodular lattices can serve as an algebraic semantics of the logic of quantum mechanics. Namely, the class of event-state systems in quantum mechanics is usually identified with the set of projection operators on a Hilbert space \({\mathbf {H}}\) and this set is in a bijective correspondence with the set of all closed linear subspaces of \({\mathbf {H}}\). However, certain doubts concerning the relevance of this representation arose when it was shown that the class of orthomodular lattices arising in this way does not generate the variety of orthomodular lattices. In other words, there exist orthomodular lattices which do not represent a physical system in the logic of quantum mechanics. The reason is that some equational properties of event-state systems are not fairly reflected by the proposed mathematical abstraction. This was the reason for alternative approaches (see, e.g., Giuntini et al. (2016, 2017)). In particular, an algebraic semantics for the logic of quantum mechanics was found among orthomodular posets and their modifications.

Orthomodular lattices have similar properties as algebraic structures used for substructural logics (see, e.g., Galatos et al. (2007)). The authors proved in Chajda and Länger (2017a, b) that every orthomodular lattice can be converted into a so-called left residuated l-groupoid. They showed in Chajda and Länger (submitted) that this result can be easily extended to a certain class of bounded lattices with a unary operation which, of course, contains the variety of orthomodular lattices. Hence, the natural question arises if this approach can be extended to ordered sets with a unary operation. For this purpose, so-called residuated operators were introduced in Chajda and Länger (2018) and several classes of ordered sets with a unary operation turned out to be operator residuated. A prominent role among these posets play the so-called pseudo-orthomodular posets which are a direct generalization of orthomodular lattices, but serve also as good candidates for an algebraic semantics of the logic of quantum mechanics. And again, there arises the natural question if the posets listed in Chajda and Länger (2018) really exhaust all possible cases.

The aim of the present paper is to provide several simple conditions under which a bounded poset with a unary operation can be organized into an operator left residuated poset. Similarly as it was done for lattices in Chajda and Länger (submitted), we ask whether these conditions are not only sufficient but also necessary. It is shown that if subsets instead of single elements are considered, then these generalized conditions characterize the class of posets which can be converted into operator residuated ones.

## 2 Adjointness of operators

*orthomodular lattice*is a bounded lattice \((L,\vee ,\wedge ,{}',0,1)\) with a unary operation \('\) which is a complementation and an antitone involution (see, e.g., Kalman 1958) satisfying the

*orthomodular law*

*left residuated lattice*(or

*integral l-groupoid*in the terminology of Galatos et al. 2007) is a bounded lattice \({\mathbf {L}}=(L,\vee ,\wedge ,\odot ,\rightarrow ,0,1)\) with two more binary operations \(\odot \) and \(\rightarrow \) satisfying

*adjointness*.

*L*(

*a*,

*b*),

*L*(

*A*,

*b*),

*L*(

*A*,

*B*), respectively. Similar expressions are treated analogously. Moreover, if \('\) is a unary operation on

*P*, then \(A':=\{x'\mid x\in A\}\).

However, as shown in Chajda and Länger (2018), if \((P,\le ,{}',0,1)\) is an orthomodular poset (or even a Boolean poset), then such operations \(\odot \) and \(\rightarrow \) need not exist. In order to avoid these complications, we study bounded ordered sets with a unary operation. We introduced in Chajda and Länger (2018) the following notion:

### Definition 2.1

*operator left residuated poset*is an ordered seventuple \({\mathbf {P}}=(P,\le ,{}',M,R,0,1)\) where \((P,\le ,{}',0,1)\) is a bounded poset with a unary operation and

*M*and

*R*are mappings from \(P^2\) to \(2^P\) satisfying the following conditions for all \(x,y,z\in P\):

- (i)
\(M(x,1)=M(1,x)=L(x)\),

- (ii)
\(M(x,y)\subseteq L(z)\) if and only if \(L(x)\subseteq R(y,z)\),

- (iii)
\(R(x,0)=L(x')\).

*operator left adjointness*. If

*M*is

*commutative*, then (ii) is called

*operator adjointness*only and \({\mathbf {P}}\) is called an

*operator residuated poset*.

In Chajda and Länger (2018), the definition contains one more condition which, however, follows from (i) and (ii) (see the following lemma).

### Lemma 2.2

### Proof

### Lemma 2.3

Let \((P,\le ,{}')\) be a poset with a unary operation and *M* and *R* defined by (3) and (4), respectively. Then, (1) implies \(M(x,y)\subseteq L(z)\Rightarrow L(x)\subseteq R(y,z)\) and (2) implies \(L(x)\subseteq R(y,z)\Rightarrow M(x,y)\subseteq L(z)\).

### Proof

### Definition 2.4

*distributive poset*is a poset \((P,\le )\) satisfying one of the following equivalent identities:

*poset with complementation*is an ordered quintuple \({\mathbf {P}}=(P,\le ,{}',0,1)\) such that \((P,\le ,0,1)\) is a bounded poset and \('\) is a unary operation on

*P*satisfying the following conditions for all \(x,y\in P\):

- (i)
\(L(x,x')\approx \{0\}\) and \(U(x,x')\approx \{1\}\),

- (ii)
\(x\le y\) implies \(y'\le x'\),

- (iii)
\((x')'\approx x\).

As mentioned in Introduction, we introduce several kinds of posets with complementation which generalize orthomodular lattices.

The poset \({\mathbf {P}}\) with complementation is called a *Boolean poset* if \((P,\le )\) is distributive. Of course, every Boolean algebra is a Boolean poset, but there are interesting examples of Boolean posets which are not lattices (see, e.g., Chajda and Länger (submitted)). In every case, Boolean posets are orthomodular posets, and hence, they can be considered as quantum structures.

*pseudo-Boolean poset*if it satisfies one of the following equivalent identities:

*M*and

*R*defined by

*pseudo-orthomodular poset*if it satisfies one of the following equivalent identities:

It is easy to see that every Boolean poset is pseudo-Boolean and every pseudo-Boolean poset is pseudo-orthomodular (cf. Chajda and Länger 2018).

### Theorem 2.5

Let \((P,\le ,{}',0,1)\) be a bounded poset with a unary operation satisfying both conditions (1) and (2) and the identity \(1'\approx 0\) and *M* and *R* defined by (3) and (4), respectively. Then, \((P,\le ,{}',M,R,0,1)\) is an operator left residuated poset.

### Proof

- (i)
\(M(x,1)\approx L(U(x,1'),1)\approx L(U(x,1'))\approx L(U(x,0))\approx L(U(x))\approx L(x)\),

\(M(1,x)\approx L(U(1,x'),x)\approx L(1,x)\approx L(x)\),

- (ii)
Follows from Lemma 2.3,

- (iii)
\(R(x,0)\approx L(U(L(0,x),x'))\approx L(U(0,x'))\approx L(U(x'))\approx L(x')\). \(\square \)

We show that the posets mentioned above are among those assumed in Theorem 2.5.

### Example 2.6

*M*is commutative.

### Lemma 2.7

Let \((P,\le ,{}')\) be a poset with a unary operation and *M* and *R* defined by (7) and (8), respectively. Then, (5) implies \(M(x,y)\subseteq L(z)\Rightarrow L(x)\subseteq R(y,z)\) and (6) implies \(L(x)\subseteq R(y,z)\Rightarrow M(x,y)\subseteq L(z)\).

### Proof

### Theorem 2.8

Let \((P,\le ,{}',0,1)\) be a bounded poset with a unary operation satisfying both conditions (5) and (6) and *M* and *R* defined by (7) and (8), respectively. Then, \((P,\le ,{}',M,R,0,1)\) is an operator residuated poset.

### Proof

- (i)
\(M(x,1)\approx L(x,1)\approx L(x)\),

\(M(1,x)\approx L(1,x)\approx L(x)\),

- (ii)
Follows from Lemma 2.7,

- (iii)
\(R(x,0)\approx L(U(0,x'))\approx L(U(x'))\approx L(x')\) .

*M*is commutative, \((P,\le ,{}',M,R,0,1)\) is an operator residuated poset. \(\square \)

Again, pseudo-Boolean and hence also Boolean posets are among those posets assumed in Theorem 2.8 (see the following example).

### Example 2.9

Combining Theorems 2.5 and 2.8 and Examples 2.6 and 2.9, we conclude

### Corollary 2.10

If \((P,\le ,{}',0,1)\) is a pseudo-Boolean poset and *M* and *R* are defined by (7) and (8), respectively, then \((P,\le ,{}',M,R,0,1)\) is an operator residuated poset. If \((P,\le ,{}',0,1)\) is a pseudo-orthomodular poset satisfying the identity \(1'\approx 0\) and *M* and *R* are defined by (3) and (4), respectively, then \((P,\le ,{}',M,R,0,1)\) is an operator left residuated poset.

It is well known that in a residuated lattice each of the operations \(\odot \) and \(\rightarrow \) determines the other one. We can prove a similar result also for the posets listed above.

### Proposition 2.11

- (i)
- (ii)

### Proof

- (i)$$\begin{aligned} L((M(y',x))')&\approx L((L(U(y',x'),x))')\\&\approx L(U(L(y,x),x'))\approx R(x,y), \\ L((R(y,x'))')&\approx L((L(U(L(x',y),y')))')\\&\approx L(U(L(U(x,y'),y)))\\&\approx L(U(x,y'),y)\\&\approx M(x,y), \end{aligned}$$
- (ii)$$\begin{aligned} L((M(y',x))')&\approx L((L(y',x))')\approx L(U(y,x'))\\&\approx R(x,y), \\ L((R(y,x'))')&\approx L((L(U(x',y')))')\approx L(U(L(x,y)))\\&\approx L(x,y)\approx M(x,y). \end{aligned}$$

## 3 A characterization of posets satisfying generalized operator residuation

The conditions (5), (6) as well as (7), (8) which are formulated for variables can be expressed also for subsets of *P* in the following way:

*M*(

*x*,

*y*) and

*R*(

*x*,

*y*) as previously. Of course, taking singletons in (9) and (10) instead of

*A*and

*B*yields (5) and (6), respectively. Hence, the new conditions and definitions include the previous ones as a particular case. Also our definition of operator adjointness can be extended to subsets of

*P*as follows:

*generalized operator left residuated poset*if it satisfies (i) and (iii) of Definition 2.1 as well as (13) and (14), i.e., if for all \(A,B,C\subseteq P\),

*generalized operator left adjointness*. If

*M*is commutative, then (15) is called

*generalized operator adjointness*only and \({\mathbf {P}}\) is called a

*generalized operator residuated poset*. It is evident that taking singletons instead of

*A*,

*B*,

*C*in generalized operator adjointness, we obtain condition (ii) from Definition 2.1. Now we are able to prove a result analogous to Lemma 2.7.

### Theorem 3.1

Let \((P,\le ,{}')\) be a poset with a unary operation and *M* and *R* defined by (11) and (12), respectively. Then, (9) implies (13) are equivalent, and (10) implies (14) are equivalent.

### Proof

Assume \(A,B,C\subseteq P\).

By Theorem 3.1, we obtain a result analogous to Theorem 2.8.

## Notes

### Acknowledgements

Open access funding provided by TU Wien (TUW). The authors thank the anonymous referees for their valuable comments which improved the quality of the paper. Support of the research by ÖAD, Project CZ 02/2019, and support of the research of the first author by IGA, Project PřF 2019 015, is gratefully acknowledged.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Human and animals rights

This article does not contain any studies with human participants or animals performed by any of the authors.

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