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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932versão On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca jan./jun. 2007
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.
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]
- Let
be a Nelson algebra. Define the derived binary terms
and
by:
Then the term reduct
is a Nelson
-algebra.
- Let
be a Nelson
-algebra. Define the derived binary term
and the derived unary term
by:
Then the term reduct
is a Nelson algebra.
- Let
be a Nelson algebra. Then
.
- 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
,
![pict](/img/revistas/ruma/v48n1/1a0497x.png)
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:
has DPC;
has EDPC; and
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
![n+1 n ⇒ x ⇒ y ≈ x ⇒ y. (E n )](/img/revistas/ruma/v48n1/1a04203x.png)
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:
has DPC;
has EDPC;
; and
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
![pict](/img/revistas/ruma/v48n1/1a04227x.png)
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
![pict](/img/revistas/ruma/v48n1/1a04332x.png)
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
![pict](/img/revistas/ruma/v48n1/1a04341x.png)
which yields (8) as claimed.
Now it is clear that
![pict](/img/revistas/ruma/v48n1/1a04342x.png)
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:
- The ternary term
is a commutative, regular TD term for
with respect to
;
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]:
![∘ ∘ ∘ ∘ x ⋅ y ≈ y ⋅ x x ⋅ y ≈ x∘ ⋅ y ∘ x ⋅ x ≈ x 1∘ ≈ 1.](/img/revistas/ruma/v48n1/1a04423x.png)
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
![x ⋅ y := e(x,1,y) x∘ := e(x,1,1) ( ) x → y := e x,e(x, y,x),1](/img/revistas/ruma/v48n1/1a04455x.png)
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
![n x ⋅ y := x * y x∘ := xn ( n )n x → y := x ⇒ (((x ⇒ y) ∧ (y ⇒ x)) * x ) .](/img/revistas/ruma/v48n1/1a04481x.png)
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
![x ⋅ y := x2 * y ∘ 2 x := x x → y := (x ⇒ (((x ⇒ y) ∧ (y ⇒ x))2 * x))2](/img/revistas/ruma/v48n1/1a04494x.png)
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
![(x ⇒ x ) ⇒ y ≈ y (x ⇒ y) ∧ y ≈ y x ∧ ~ (x ∧ ~ y) ≈ x ∧ (x ⇒ y) x ⇒ (y ∧ z) ≈ (x ⇒ y) ∧ (x ⇒ z ) x ⇒ y ≈ ~ y ⇒ ~ x ( ) ( ) x ⇒ x ⇒ (y ⇒ (y ⇒ z)) ≈ (x ∧ y ) ⇒ (x ∧ y) ⇒ z ~(~ x ∧ y) ⇒ (x ⇒ y) ≈ x ⇒ y x ∧ (x ∨ y) ≈ x x ∧ (y ∨ z) ≈ (z ∧ x) ∨ (z ∧ y) (x ∧ ~ x) ∨ (y ∨ ~ y) ≈ x ∧ ~ x](/img/revistas/ruma/v48n1/1a04504x.png)
where the derived nullary term is defined as
![1 := 0 ⇒ 0, (1 ) def](/img/revistas/ruma/v48n1/1a04506x.png)
the derived unary term is defined as in (2), and the derived binary term
is defined by
![x ∨ y := ~ (~ x ∧ ~ y). (∨def)](/img/revistas/ruma/v48n1/1a04509x.png)
Brignole established the following result:
Theorem 4.1. [10]
- Let
be a Nelson algebra and define the derived binary term
as in (⇒def). Then the term reduct
is a member of
.
- 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.
- Let
be a Nelson algebra. Then
.
- 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