# Left residuated lattices induced by lattices with a unary operation

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## Abstract

In a previous paper, the authors defined two binary term operations in orthomodular lattices such that an orthomodular lattice can be organized by means of them into a left residuated lattice. It is a natural question if these operations serve in this way also for more general lattices than the orthomodular ones. In our present paper, we involve two conditions formulated as simple identities in two variables under which this is really the case. Hence, we obtain a variety of lattices with a unary operation which contains exactly those lattices with a unary operation which can be converted into a left residuated lattice by use of the above mentioned operations. It turns out that every lattice in this variety is in fact a bounded one and the unary operation is a complementation. Finally, we use a similar technique by using simpler terms and identities motivated by Boolean algebras.

## Keywords

Left adjointness Left residuated lattice Orthomodular lattice Variety of lattices Weakly orthomodular lattice Dually weakly orthomodular lattice ComplementationIt was recently shown by the authors in Chajda and Länger (2017a, b) that every orthomodular lattice \(\mathbf L\) can be converted into a left residuated lattice, i.e., there exist binary operations \(\otimes \) and \(\rightarrow \) such that \(x\otimes 1\approx 1\otimes x\approx x\) and the so-called left adjointness is satisfied by \(\otimes \) and \(\rightarrow \). It is known that orthomodular lattices were used in the 1930s by Husimi (1937) and Birkhoff and Von Neumann (1936) as an algebraic axiomatization of the semantics of the logic of quantum mechanics. However, it was recognized later that this axiomatization is not precise since e.g., joins of elements in these ordered sets need not exist. This was the reason why so-called orthomodular posets were introduced. Because the precise algebraic axiomatization is not known until now, it motivates us to study more general lattices than orthomodular ones which, however, formally satisfy the law of orthomodularity which is the keystone in this theory. However, we do not demand additional conditions on the unary operation, i.e., neither complementarity nor antitone involution is supposed. The aim of this paper is to get reasonable sufficient conditions under which this lattice can be converted into a left residuated lattice.

We start with the following definition.

### Definition 1

*left residuated lattice*[see Chajda and Länger (2017a)] if

- (i)
\((L,\vee ,\wedge ,0,1)\) is a bounded lattice,

- (ii)
\(x\otimes 1\approx 1\otimes x\approx x\),

- (iii)
for arbitrary \(x,y,z\in L\), \(x\otimes y\le z\) if and only if \(x\le y\rightarrow z\).

*left adjointness*. If, moreover, \(\otimes \) is commutative and associative then \(\mathbf L\) is called a

*residuated lattice*. Given a left residuated lattice, we define \(x^{\prime }:=x\rightarrow 0\) for all \(x\in L\).

Throughout this paper, we assume that all lattices are non-empty.

The following concepts were introduced by the authors in Chajda and Länger (2018):

*weakly orthomodular*if

*dually weakly orthomodular*if

Let us note that every orthomodular lattice [see e.g., Beran (1985)] is both weakly and dually weakly orthomodular and that its unary operation is both an antitone involution and a complementation. However, when defining weak orthomodularity or dually weak orthomodularity, we neither ask the unary operation to be an antitone involution nor a complementation. Moreover, we do not assume the existence of a least or greatest element. The following was also shown in Chajda and Länger (2018).

### Remark 2

*complementation*, i.e., the identities \(x\vee x^{\prime }\approx 1\) and \(x\wedge x^{\prime }\approx 0\) are satisfied. Contrary to the case of orthomodular lattices, this complementation need not be an antitone involution.

The following examples are taken from Bonzio and Chajda (2017) and Chajda and Länger (2018).

### Example 3

### Example 4

- (3)
\(x\le ((x\vee y^{\prime })\wedge y)\vee y^{\prime }\),

- (4)
\(x\ge ((x\wedge y)\vee y^{\prime })\wedge y\).

### Lemma 5

- (i)
Assume \(\mathbf L\) to satisfy identity (3). Then, \(\mathbf L\) has a greatest element 1 and satisfies the identity \(x\vee x^{\prime }\approx 1\). If, moreover, \(\mathbf L\) has a smallest element 0 then it satisfies the identity \(0'\approx 1\).

- (ii)
Assume \(\mathbf L\) to satisfy identity (4). Then, \(\mathbf L\) has a smallest element 0 and satisfies the identity \(x\wedge x^{\prime }\approx 0\). If, moreover, \(\mathbf L\) has a greatest element 1 then it satisfies the identity \(1'\approx 0\).

- (iii)
Assume \(\mathbf L\) to satisfy both identities (3) and (4). Then, \(\mathbf L\) is bounded, \('\) a complementation of \(\mathbf L\) and, moreover, \('\) is

*switching*, i.e., the identities \(0'\approx 1\) and \(1'\approx 0\) are satisfied.

### Proof

- (i)Since \((x\vee y^{\prime })\wedge y\le y\), we have \(x\le y\vee y^{\prime }\) for all \(x,y\in L\), and hence, \(\mathbf L\) has a greatest element 1 and satisfies the identity \(x\vee x^{\prime }\approx 1\). If, moreover, \(\mathbf L\) has a smallest element 0 thenfor all \(x\in L\), i.e., \(\mathbf L\) satisfies the identity \(0'\approx 1\).$$\begin{aligned} x=x\wedge (((x\vee 0^{\prime })\wedge 0)\vee 0^{\prime })=x\wedge 0^{\prime }\le 0^{\prime } \end{aligned}$$
- (ii)Since \(y'\le (x\wedge y)\vee y'\), we have \(y'\wedge y\le x\) for all \(x,y\in L\), and hence, \(\mathbf L\) has a smallest element 0 and satisfies the identity \(x\wedge x'\approx 0\). If, moreover, \(\mathbf L\) has a greatest element 1 thenfor all \(x\in L\), i.e., \(\mathbf L\) satisfies the identity \(1'\approx 0\).$$\begin{aligned} 1'\le x\vee 1'=x\vee (((x\wedge 1)\vee 1')\wedge 1)=x \end{aligned}$$
- (iii)
follows from (i) and (ii).\(\square \)

In the following, let \(\mathcal V\) denote the variety of all algebras \((L,\vee ,\wedge ,{}',0,1)\) of type (2, 2, 1, 0, 0) such that \((L,\vee ,\wedge ,0,1)\) is a bounded lattice and \((L,\vee ,\wedge ,{}')\) satisfies both identities (3) and (4).

If \((L,\vee ,\wedge )\) is a lattice having a smallest element 0, but no greatest element and \(x':=0\) for all \(x\in L\) then \((L,\vee ,\wedge ,{}')\) satisfies (4), but not (3). If, conversely, \((L,\vee ,\wedge )\) is a lattice having a greatest element 1, but no smallest element and \(x':=1\) for all \(x\in L\) then \((L,\vee ,\wedge ,{}')\) satisfies (3), but not (4). This shows that (3) and (4) are independent.

- (3)
\(x\wedge ((x\otimes y)\vee y')\approx x\) or \(x\wedge (y\rightarrow (x\vee y'))\approx x\),

- (4)
\(x\vee ((x\wedge y)\otimes y)\approx x\) or \(x\vee ((y\rightarrow x)\wedge y)\approx x\),

Lattices with a unary operation satisfying (3) or (4) are not curious.

### Remark 6

The following examples show two typical members of \(\mathcal V\). The first one is even modular.

### Example 7

### Example 8

In the following, we investigate connections between the identities (3) and (4) and left adjointness.

### Lemma 9

### Proof

- (i)
\(a\le ((a\vee b')\wedge b)\vee b'=(b\wedge (a\vee b')\wedge b)\vee b'\le (b\wedge c)\vee b'=b\rightarrow c\) and

- (ii)
\(a\otimes b=(a\vee b')\wedge b\le ((b\wedge c)\vee b'\vee b')\wedge b=((c\wedge b)\vee b')\wedge b\le c\).\(\square \)

Using these results, we can characterize the class of all lattices with a unary operation which can be organized into left residuated lattices via the operations \(\otimes \) and \(\rightarrow \) defined by (1) and (2), respectively.

### Theorem 10

Let \((L,\vee ,\wedge ,{}')\) be a lattice with a unary operation and \(\otimes \) and \(\rightarrow \) defined by (1) and (2), respectively. Then, \((L,\vee ,\wedge ,\otimes ,\rightarrow ,0,1)\) is a left residuated lattice satisfying \(x\rightarrow 0\approx x'\) if and only if \((L,\vee ,\wedge ,{}',0,1)\in \mathcal V\).

### Proof

The importance of this approach is that residuated lattices serve as an algebraic semantics of certain substructural logics. Hence, this is the link between orthomodular, weakly orthomodular and dually weakly orthomodular lattices on one side and kinds of fuzzy logics on the other side.

In the following, we investigate connections between the identities (3) and (4) and (dually) weak orthomodularity.

### Lemma 11

### Proof

- (i)Assume \(\mathbf L\) to satisfy the identity \(x''\approx x\). If \(\mathbf L\) satisfies identity (3) and \(a\le b\) thenIf, conversely, \(\mathbf L\) is weakly orthomodular then$$\begin{aligned} b\le ((b\vee a'')\wedge a')\vee a''=a\vee (b\wedge a')\le b. \end{aligned}$$$$\begin{aligned} a\le a\vee b'=b'\vee ((a\vee b')\wedge b'')=((a\vee b')\wedge b)\vee b'. \end{aligned}$$
- (ii)If \(\mathbf L\) satisfies identity (4) and \(a\le b\) thenIf, conversely, \(\mathbf L\) is dually weakly orthomodular then$$\begin{aligned} a\le b\wedge (a\vee b')=((a\wedge b)\vee b')\wedge b\le a. \end{aligned}$$$$\begin{aligned} ((a\wedge b)\vee b')\wedge b=b\wedge ((a\wedge b)\vee b')=a\wedge b\le a.\square \end{aligned}$$

If the identity \(x''\approx x\) is not satisfied then identity (3) need not be equivalent to weak orthomodularity as the following example shows:

### Example 12

Summarizing, we obtain the following result:

### Theorem 13

If \(\mathbf L=(L,\vee ,\wedge ,{}')\) is a lattice with a unary operation satisfying the identity \(x''\approx x\) and being both weakly and dually weakly orthomodular then \((L,\vee ,\wedge ,\otimes ,\rightarrow ,0,1)\) is a left residuated lattice.

### Example 14

The variety \(\mathcal V\) has several important subvarieties, e.g., the variety of bounded complemented modular lattices (due to Remark 6), the variety \(\mathcal O\) of orthomodular lattices and, of course, the variety of Boolean algebras. Since \(\mathcal V\supseteq \mathcal O\), the variety \(\mathcal V\) is residually large since for every infinite cardinal *k* there exists a subdirectly irreducible orthomodular lattice of cardinality *k* [(cf. Chajda and Länger (2020)].

In the lattice from Example 14, we have \(c\vee c'\ne 1\). We show that this is just the condition needed to prove left adjointness provided the operation \(\rightarrow \) is defined by (2).

### Theorem 15

Let \(\mathbf L=(L,\vee ,\wedge ,{}',0,1)\) be a bounded lattice with a unary operation and \(a\in L\) with \(a\vee a'\ne 1\). Then there does not exist a binary operation \(\otimes \) on *L* such that \((L,\vee ,\wedge ,\otimes ,\rightarrow ,0,1)\) is a left residuated lattice where \(\rightarrow \) is defined by (2).

### Proof

Assume there exists some binary operation \(\otimes \) on *L* such that \((L,\vee ,\wedge ,\otimes ,\rightarrow ,0,1)\) is a left residuated lattice. Because of \(1\otimes a\le a\) we conclude \(1\le a\rightarrow a=(a\wedge a)\vee a'=a\vee a'\), a contradiction.\(\square \)

However, also in the case that *L* contains an element *a* satisfying \(a\vee a'\ne 1\), \(\mathbf L\) may be converted into a left residuated lattice if the operations \(\otimes \) and \(\rightarrow \) are defined in an appropriate way, see the following example.

### Example 16

- (7)
\(x\le (x\wedge y)\vee y'\),

- (8)
\(x\ge (x\vee y')\wedge y\).

### Lemma 17

### Proof

- (i)
\(a\le (a\wedge b)\vee b'=b'\vee (a\wedge b)\le b'\vee c=b\rightarrow c\) and

- (ii)
\(a\otimes b=a\wedge b\le (b'\vee c)\wedge b=(c\vee b')\wedge b\le c\).\(\square \)

From Lemma 17, we conclude

### Theorem 18

If \(\mathbf L=(L,\vee ,\wedge ,{}',0,1)\) is a bounded lattice with a unary operation satisfying both identities (7) and (8) and \(\otimes \) and \(\rightarrow \) are defined by (5) and (6), respectively, then \((L,\vee ,\wedge ,\otimes ,\rightarrow ,0,1)\) is a residuated lattice.

Using these results, we can characterize the class of all bounded lattices with a unary operation which can be organized into residuated lattices via the operations \(\otimes \) and \(\rightarrow \) defined by (5) and (6), respectively.

### Theorem 19

Let \((L,\vee ,\wedge ,{}',0,1)\) be a bounded lattice with a unary operation and \(\otimes \) and \(\rightarrow \) defined by (5) and (6), respectively. Then, \((L,\vee ,\wedge ,\otimes ,\rightarrow ,0,1)\) is a residuated lattice satisfying \(x\rightarrow 0\approx x'\) if and only if \((L,\vee ,\wedge ,{}')\) satisfies both identities (7) and (8).

### Proof

Let \(a,b\in L\). If \((L,\vee ,\wedge ,\otimes ,\rightarrow ,0,1)\) is a residuated lattice satisfying \(x\rightarrow 0\approx x'\) then \((L,\vee ,\wedge ,0,1)\) is a bounded lattice, since \(a\otimes b\le a\wedge b\) we have \(a\le b\rightarrow (a\wedge b)=(a\wedge b)\vee b'\), i.e., identity (7) is satisfied, and because of \(b'\vee a\le b\rightarrow a\) we have \((a\vee b')\wedge b=(b'\vee a)\otimes b\le a\), i.e., identity (8) is satisfied. Conversely, if \((L,\vee ,\wedge ,{}')\) satisfies both identities (7) and (8) then \((L,\vee ,\wedge ,\otimes ,\rightarrow ,0,1)\) is a residuated lattice satisfying \(x\rightarrow 0\approx x'\) according to Lemma 17.\(\square \)

### Corollary 20

The class \(\mathcal C\) of bounded lattices with a unary operation \('\) satisfying the identity \(x''\approx x\) which become residuated lattices with respect to the operations \(\otimes \) and \(\rightarrow \) defined by (5) and (6) coincides with the variety of Boolean algebras.

### Proof

Recall that an *algebra*\(\mathbf A\) is called *congruence distributive* if its congruence lattice is distributive, and \(\mathbf A\) is called *congruence permutable* if \(\Theta \circ \Phi =\Phi \circ \Theta \) for all congruences \(\Theta ,\Phi \) on \(\mathbf A\). A *variety* is called congruence distributive (congruence permutable) if any of its members has the corresponding property. For further information concerning these notions cf. Chajda et al. (2012).

Then, we can prove even more.

### Theorem 21

The variety \(\mathcal W\) is congruence permutable.

### Proof

*p*denote the term defined by

*p*is a Maltsev term. Hence, \(\mathcal W\) is congruence permutable.\(\square \)

## Notes

### Acknowledgements

Open access funding provided by TU Wien (TUW). The authors thank the anonymous referees for their useful comments which improved the quality of the paper.

### Funding

This study was funded by ÖAD, Project CZ 02/2019, and, concerning the first author, by IGA, Project PřF 2019 015.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Human and animal rights

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

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