Characterisations of Nelson algebras

M. Spinks and R. Veroff

Abstract. Nelson algebras arise naturally in algebraic logic as the algebraic models of Nelson's constructive logic with strong negation. This note gives two characterisations of the variety of Nelson algebras up to term equivalence, together with a characterisation of the finite Nelson algebras up to polynomial equivalence. The results answer a question of Blok and Pigozzi and clarify some earlier work of Brignole and Monteiro.

Key words and phrases. Nelson algebra, residuated lattice, BCK-algebra, equationally definable principal congruences

The first author would like to thank Nick Galatos for several helpful conversations about residuated lattices.
The final version of this paper was prepared while the first author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and the Department are gratefully acknowledged.

1. Introduction

Recall from the theory of distributive lattices [2, Chapter XI] that a De Morgan algebra is an algebra of type where is a bounded distributive lattice and for all , , and .

A Nelson algebra (also -lattice or quasi-pseudo-Boolean algebra in the literature) is an algebra of type such that the following conditions are satisfied for all  [26, Section 0]:

(N1)   is a De Morgan algebra with lattice ordering ;

(N2)   The relation  defined for all by if and only if is a quasiordering (reflexive and transitive relation) on ;

(N3)   if and only if ;

(N4)   if and only if and ;

(N5)   and implies ;

(N6)   and implies ;

(N7)   and ;

(N8)   and ;

(N9)   .

The class  of all Nelson algebras is a variety [11], which arises naturally in algebraic logic as the equivalent quasivariety semantics (in the sense of [4]) of Nelson's constructive logic with strong negation [25, Chapter XII]. For studies of Nelson algebras see in particular Sendlewski [26], Vakarelov [31], and Rasiowa [25, Chapter V].

A commutative, integral residuated lattice is an algebra of type , where: (i) is a lattice with lattice ordering  such that for all ; (ii) is a commutative monoid; and (iii) for all , if and only if . By Blount and Tsinakis [9, Proposition 4.1] the class of all commutative, integral residuated lattices is a variety. An -algebra is a commutative, integral residuated lattice with distinguished least element . The variety of all -algebras arises naturally in algebraic logic in connection with the study of substructural logics; see [19, 21, 22, 23] for details.

An -algebra  is said to be 3-potent when for all , distributive when its lattice reduct is distributive, and classical when for all . Rewriting as for all , classicality expresses the law of double negation in algebraic form. A Nelson -algebra is a 3-potent, distributive classical -algebra such that for all .

The following description (to within term equivalence) of the variety of Nelson algebras was obtained by the authors in [29, 30].

Theorem 1.1. [29, Theorem 1.1]

1. Let be a Nelson algebra. Define the derived binary terms  and  by:

   

   Then the term reduct is a Nelson -algebra.

2. Let be a Nelson -algebra. Define the derived binary term  and the derived unary term  by:

   

   Then the term reduct is a Nelson algebra.

3. Let be a Nelson algebra. Then .
4. Let be a Nelson -algebra. Then .

Hence the varieties of Nelson algebras and Nelson -algebras are term equivalent.

In this note we give several further characterisations of Nelson algebras, all of which may be understood as corollaries of Theorem 1.1. We shall make implicit use of Theorem 1.1 without further reference throughout the paper.

A BCK-algebra is a -subreduct of a commutative, integral residuated lattice [32, Theorem 5.6]; for an equivalent quasi-equational definition, see Section 2. We show in Section 2 that every finite Nelson algebra  is polynomially equivalent to its own BCK-algebra term reduct .

A pseudo-interior algebra is a hybrid of a (topological) interior algebra and a residuated partially ordered monoid; for a precise definition, see Section 3 below. We prove in Section 3 that the variety of Nelson algebras is term equivalent to a congruence permutable variety of pseudo-interior algebras with compatible operations. We obtain this result as a byproduct of the solution to a problem of Blok and Pigozzi [5].

A lower BCK-semilattice is the conjunction of a meet semilattice with a BCK-algebra such that the natural partial orderings on both algebras coincide; for a formal definition see Section 4 below. We verify in Section 4 that the variety of Nelson algebras is term equivalent to a variety of bounded BCK-semilattices. The result clarifies earlier work on the axiomatics of Nelson algebras due to Brignole [10].

2. Finite Nelson algebras as BCK-algebras

A BCK-algebra is an algebra of type such that the following identities and quasi-identity are satisfied:

(1)

By Wroński [34] the quasivariety of all BCK-algebras is not a variety. BCK-algebras have been considered extensively in the literature; for surveys, see Iséki and Tanaka [17] or Cornish [13]. Here we simply recall that for any BCK-algebra , the relation on  defined for all by if and only if is a partial ordering, which has the property that for any ,

A non-empty subset  of a BCK-algebra  is said to be a BCK-filter if and implies for all . Also, a non-empty subset of an -algebra  is said to be an -filter if: (i) and implies ; and (ii) implies for all . It is easy to see that a non-empty subset of an -algebra  is an -filter if and only if it is a BCK-filter of the BCK-algebra reduct of  [19, p. 12].

Let  be an -algebra [resp. BCK-algebra]. It is well known and easy to see that every congruence  on  [resp. congruence  on  such that is a BCK-algebra] is of the form for some -filter [resp. BCK-filter] , where for all , if and only if (put ). See [19, Proposition 1.3] for the case of -algebras and [7, Proposition 1] for the case of BCK-algebras.

For any , let . In view of the preceding discussion, we have

Lemma 2.1. For any -algebra , . In particular, if , then .

Recall from [3] that a class  of similar algebras has definable principal congruences (DPC) if and only if there exists a formula in the first-order language of (whose only free variables are ) such that for any and , if and only if . When can be taken as a conjunction (viz., finite set) of equations, then is said to have equationally definable principal congruences (EDPC) [15].

For any integer , consider the unary -terms defined recursively by and when . Given , an element of an -algebra  is said to be -potent if . is said to be -potent if it satisfies an identity of the form

(E)

Clearly the class of all -algebras satisfying (E) is equationally definable.

Theorem 2.2. [18, Theorem 2.1] For a variety  of  -algebras the following conditions are equivalent:

1. has DPC;
2. has EDPC; and
3. for some .

A ternary term is a ternary deductive (TD) term for an algebra  if , and, for all , if [5, Definition 2.1]. is said to be a ternary deductive (TD) term for a class  of similar algebras if it is a TD term for every member of . By [5, Theorem 2.5], for any if and only if , whence  has EDPC. A TD term for an algebra  is said to be commutative if in addition . A TD term for a class  of similar algebras is said to be commutative if it is commutative for every member of  [5, Definition 3.1].

For any integer , consider the binary -terms defined recursively by and when . Given , a BCK-algebra is said to be -potent if it satisfies an identity of the form

By Cornish [12, Theorem 1.4] the class of all BCK-algebras satisfying (2) is a variety.

Theorem 2.3. [8, Theorem 4.2] For , the following conditions are equivalent for a variety  of BCK-algebras:

1. has DPC;
2. has EDPC;
3. ; and
4. is a commutative TD term for .

Let  be a variety of -algebras. Suppose for some and let . By [7, Proposition 13, Lemma 14], if and only if if and only if , whence is -potent. Since and is a variety of BCK-algebras, . By Lemma 2.1, therefore, and hence

where denotes the commutative TD term of Theorem 2.3. Of course, . Thus is a commutative TD term for .

Conversely, suppose is a commutative TD term for . Let . By [5, Theorem 2.3, Corollary 2.4], , which is to say  satisfies

(4)

Therefore . But by [7, Proposition 13], a BCK-algebra satisfies (4) if and only if it satisfies (2). Hence is -potent. By the remarks following Theorem 2.3 we infer that is -potent, whence . We have established

Proposition 2.4. For , is a commutative TD term for a variety  of  -algebras if and only if .

In [29, Proposition 3.2] the authors showed that the variety of Nelson algebras satisfies the identity , where  denotes the derived binary term defined as in (⇒_def). See also Viglizzo [33, Chapter 1]. Since the variety of all Nelson -algebras is 3-potent, we have

Corollary 2.5. [28, Theorem 3.3, Remark 3.5] is a commutative TD term for the variety of Nelson algebras, where  denotes the derived binary term defined as in (⇒_def).

Proof. By Blok and Pigozzi [5, Theorem 2.3(iii)] the property of being a commutative TD term for a variety can be characterised solely by equations, so the result follows from the remarks preceding the corollary and Proposition 2.4. □

Next, recall the following classic result from the theory of -algebras.

Proposition 2.6. [16, Theorem 2] The variety of -algebras is arithmetical. A Mal'cev term for is .

From Proposition 2.6 we infer

Theorem 2.7. [28, Theorem 4.4] The variety of Nelson algebras is arithmetical. A Mal'cev term for is , where  denotes the derived binary term defined as in (⇒_def).

Let  be a finite -algebra. Because the monoid reduct of  is finite, must be -potent for some . See also Cornish [13, p. 419]. By the remarks following Theorem 2.3, we infer that is -potent and hence that . We therefore have

Theorem 2.8. Every finite -algebra  is polynomially equivalent to its BCK-algebra reduct .

Proof. The result follows Lemma 2.1, Proposition 2.6 and a result due to Pixley [24, Theorem 1], which asserts that if is a finite algebra in an arithmetical variety and is a function preserving congruences on  then is a polynomial of . □

Corollary 2.9. Every finite Nelson algebra  is polynomially equivalent to its BCK-algebra term reduct .

The class of all -term reducts of Nelson algebras is strictly contained within the variety of all 3-potent BCK-algebras. In particular, it can be shown that satisfies the identity

(Commutative) BCK-algebras satisfying the identity (L) have been studied extensively by Dvurečenskij and his collaborators in a series of papers beginning with [14].

It is easy to see that 3-potent BCK-algebras need not satisfy (L) in general. Hence, Corollary 2.9 prompts the following

Problem 2.10. Characterise the -reducts of Nelson algebras.

3. Nelson algebras as pseudo-interior algebras

A BCI-monoid is an algebra where: (i) is a semilattice; (ii) is a commutative monoid; and for all , both (iii) implies and ; and (iv) if and only if [1, Section 2]. An integral BCI-monoid is a BCI-monoid  satisfying for all . By [1, Proposition 2.8] the class of all (integral) BCI-monoids is equationally definable. For a recent study of BCI-monoids, see Olson [20].

For any integer , consider again the unary -terms defined recursively by and when . A unary operation  on an integral BCI-monoid  is said to be compatible if for any there is an such that [1, Section 2]. An -ary operation on  is said to be compatible if is compatible for any and . An integral BCI-monoid with compatible operations is an algebra such that is an integral BCI-monoid and any is compatible. By Aglianó [1, Remarks following Proposition 2.13] is an integral BCI-monoid with compatible operations if and only if is an integral BCI-monoid and the congruences on  are determined by  in the sense that .

Lemma 3.1. The variety of commutative, integral residuated lattices satisfies the identity:

(5)

Proof. Let and let . To establish (5), note first that satisfies the identities

Indeed, from and (2) we have , which yields (6). Similarly, from and (3) we have , which gives (7).

Next, note that satisfies the identity

(8)

Indeed, from the theory of residuated lattices [9, Lemma 3.2] we have that satisfies the identity

(9)

But then

which yields (8) as claimed.

Now it is clear that

which gives (5) as desired. □

It is well known and easy to see that for any , implies and for all . The -reduct of any commutative, integral residuated lattice is thus an integral BCI-monoid. Moreover, commutativity of the monoid operation  together with Lemma 3.1 guarantees that the lattice join  is compatible with . Hence we have

Lemma 3.2. The variety of commutative, integral residuated lattices, hence -algebras, is a variety of integral BCI monoids with compatible operations.

A TD term for an algebra  with a constant term  is said to be regular (for ) with respect to  if for all [5, Definition 4.1]. A TD term for a variety with a constant term  is said to be regular (for ) with respect to  if it is regular with respect to  for every member of .

Theorem 3.3. [1, Theorem 3.1, Corollary 3.2] For , the following conditions are equivalent for a variety  of integral BCI-monoids with compatible operations:

1. The ternary term is a commutative, regular TD term for  with respect to ;
2. has EDPC: for any and ,

Let  be a variety of -algebras. Observe that for any and , the statement

(10)

is equivalent to its corresponding statement about -filters, viz.:

(11)

where  denotes the principal filter generated by . We claim that (11) is equivalent to the assertion . So assume is -potent. We have if and only if for some  if and only if (because , for ). Conversely, suppose (11) holds. Clearly, -potency is equivalent to , which in view of (11) reduces to , which statement is true.

From the preceding discussion it follows that if and only if (10) holds for any and . Combining Lemma 3.2 with Theorem 3.3 therefore yields

Proposition 3.4. For , is a commutative, regular TD term with respect to  for a variety  of  -algebras if and only if .

In [5, Problem 7.4] Blok and Pigozzi asked whether the variety of Nelson algebras has a commutative, regular TD term, or even a TD term; for a discussion and references, see Spinks [28]. The following corollary, in conjunction with Corollary 2.5, completely resolves this question. But first, for a term in the language of Nelson algebras, let abbreviate , where  denotes the derived binary term defined as in (∗_def).

Corollary 3.5. is a commutative, regular TD term with respect to  for the variety of Nelson algebras, where  and  denote the derived binary terms defined as in (⇒_def) and (∗_def) respectively.

Proof. Since the property of being a commutative, regular TD term for a variety can be characterised solely by equations (by [5, Theorem 2.3(iii)] and [5, Corollary 4.2(i)]), the result follows from 3-potency and Proposition 3.4. □

Let be a semigroup with a constant  that acts as a left identity for . A unary operation on   is said to be a pseudo-interior operation on if the following identities are satisfied [6, Definition 2.1]:

Given a semigroup  with left identity  and pseudo-interior operation , the inverse right-divisibility ordering on  is the partial ordering  defined for all by if and only if there exists such that [6, Lemma 2.3].

An algebra of type is said to be a pseudo-interior algebra if [6, Definition 2.6]: (i) is a semigroup with left identity ; (ii) is a pseudo-interior operation on ; and (iii) is an open left residuation on in the sense that for all , and moreover if and only if for all . An algebra is said to be a pseudo-interior algebra with compatible operations if is a pseudo-interior algebra and the congruences on  are determined by  in the sense that [6, Definition 2.7, Corollary 2.17]. By [6, Theorem 3.1] the class of all pseudo-interior algebras, with or without compatible operations, is equationally definable.

Theorem 3.6. [6, Theorem 4.1, Corollary 4.2] A variety has a commutative, regular TD term if and only if it is term equivalent to a variety of pseudo-interior algebras with compatible operations. If is a commutative, regular TD term for  with respect to , then

define terms realising the pseudo-interior operations , , and  on any such that all the fundamental operations of  are compatible with .

Let  be a variety of -algebras. If for some , then  is term equivalent to a congruence permutable variety of pseudo-interior algebras with compatible operations by Proposition 2.6, Proposition 3.4, and Theorem 3.6. Conversely, if is term equivalent to a congruence permutable variety of pseudo-interior algebras with compatible operations, then for some by Theorem 3.6, EDPC, and Theorem 2.2. Therefore we have

Theorem 3.7. A variety  of -algebras is term equivalent to a congruence permutable variety of pseudo-interior algebras with compatible operations if and only if   for some . If  , then for any , terms realising the pseudo-interior operations , , and  on  are defined by

Proof. It remains only to establish the second assertion of the theorem. When is -potent, the terms realising the pseudo-interior operations on any member of  may be obtained by instantiating Theorem 3.6 with the TD term of Proposition 3.4 and simplifying the resulting expressions for , and using the now well-developed arithmetic of commutative, integral residuated lattices [9, 19]. □

Since the variety of all Nelson -algebras is 3-potent, from Theorem 3.7 we have

Corollary 3.8. The variety of Nelson algebras is term equivalent to a congruence permutable variety of pseudo-interior algebras with compatible operations. For any , terms realising the pseudo-interior operations , , and  on  are defined by

where the derived binary terms  and  are defined as in (⇒_def) and (∗_def) respectively.

4. Nelson algebras as bounded BCK-semilattices

In 1963 D. Brignole resolved a problem, posed by A. Monteiro, that asked whether Nelson algebras could be defined in terms of the connectives , and the constant . See [10] or [33, Chapter 1, p. 21]. Let  (for Brignole) denote the variety of all algebras of type axiomatised by the following collection of identities

where the derived nullary term  is defined as

the derived unary term  is defined as in (2), and the derived binary term  is defined by

Brignole established the following result:

Theorem 4.1. [10]

1. Let be a Nelson algebra and define the derived binary term  as in (⇒_def). Then the term reduct is a member of  .
2. Let be a member of  . Define the derived binary terms  and  as in (4) and (2) respectively, the derived unary term  as in (2) and the derived nullary term  as in (4). Then the term reduct is a Nelson algebra.
3. Let be a Nelson algebra. Then .
4. Let be a member of  . Then .

Hence the variety of Nelson algebras and the variety  are term equivalent.

A lower BCK-semilattice is an algebra where [27, Lemma 1.6.24]: (i) is a BCK-algebra; (ii) is a lower semilattice; and (iii) for all , if and only if , where  and  denote the semilattice and BCK-algebra partial orderings respectively. Lower BCK-semilattices have been studied in particular by Idziak [16]. A bounded lower BCK-semilattice is a lower BCK-semilattice with distinguished least element . A (bounded) lower BCK-semilattice is said to be -potent if its BCK-algebra reduct is -potent.

Let  denote the variety of all algebras having type axiomatised by the identities defining given above together with the identity . It is clear that is term equivalent to  and therefore also to both  and . The following result illuminates Brignole's description of Nelson algebras given in Theorem 4.1 above.

Theorem 4.2. The variety of Nelson algebras is term equivalent to a variety of bounded 3-potent BCK-semilattices.

Proof. It suffices to show any is a bounded 3-potent lower BCK-semilattice. By [29, Theorem 3.7] the -term reducts of members of are 3-potent BCK-algebras. Hence is a 3-potent BCK-algebra. Of course, is a lower semilattice. By Rasiowa [25, Theorem V.1.1], if and only if for all for any Nelson algebra , where  and  denote the lattice and BCK-algebra partial orders respectively. Hence the semilattice partial order and the BCK-algebra partial order coincide on , and  is a 3-potent lower BCK-semilattice. Finally, is clearly the least element of , whence is a bounded 3-potent lower BCK-semilattice. □

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Matthew Spinks
Department of Education,
University of Cagliari,
Cagliari 09123 Italy
mspinksau@yahoo.com.au

Robert Veroff
Department of Computer Science,
University of New Mexico,
Albuquerque, NM 87131 USA
veroff@cs.unm.edu

Recibido: 3 de mayo de 2005
Aceptado: 28 de marzo de 2007