Coproducts of Kleene algebras

Abstract.We develop a Priestley-style duality theory for different classes of algebras having a bilattice reduct. A similar investigation has already been realized by B. Mobasher, D. Pigozzi, G. Slutzki and G. Voutsadakis, but only from an abstract category-theoretic point of view. In the present work we are instead interested in a concrete study of the topological spaces that correspond to bilattices and some related algebras that are obtained through expansions of the algebraic language.Keywords: bilattices, Priestley duality theory, bilattices with conflation, bilattices with implication, Brouwerian bilattices. IntroductionBilattices are algebraic structures introduced in 1988 by Matthew Ginsberg [14] as a uniform framework for inference in Artificial Intelligence. Since then they have found a variety of applications, sometimes in quite different areas from the original one. The interest in bilattices has thus different sources: among others, computer science and A.I. (see especially the works of Ginsberg, Arieli and Avron), logic programming (Fitting), lattice theory and algebra [16] and, more recently, algebraic logic [4,5,24]. An up-todate review of the applications of this formalism and also of the motivation behind its study can be found in the dissertation [24].In the present work we develop a Priestley-style duality theory for bilattices and some related algebras that are obtained by adding new operations to the basic algebraic language of bilattices. The main idea guiding our work is that it is possible to view bounded bilattices and related algebras as bounded lattices having two extra constants that satisfy certain properties. This approach will enable us to apply known results on duality theory for different classes of lattices to the study of bilattices and other algebras having a bilattice reduct.A duality theory for bilattices has already been introduced by Mobasher et al. in [16]. However, while the point of view of [16] is abstract and category-theoretic, in the present paper instead we are interested in a concrete study of the topological spaces that correspond to bilattices and related algebras. We will briefly review the results of [16] and discuss the differences between their approach and ours in Sections 1.2 and 1.6.The rest of the paper is organized as follows. In Section 1 we introduce some definitions and algebraic results on bilattices and language expansions thereof (bilattices with a dual negation operation, bilattices with implication) that will be needed to develop our duality theory. There are no new results here but it is included for ease of reference. Section 1.6 is crucial: it is here where we introduce the fundamental point of view with which we approach the topic. It is concluded with an overview of the various algebras that are considered in this paper.In Section 2 we recall some known results on duality theories for De Morgan algebras and N4-lattices, on which we will base our treatment of various classes of bilattices in Section 3. We start in Section 3.1 with a duali...

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“…The duality theory for bounded De Morgan lattices 4 was developed in [7,8], to which we refer for all proofs and further details.…”

Section: De Morgan and Kleene Algebrasmentioning

confidence: 99%

Priestley Duality for Bilattices

Jung

Rivieccio

2012

Stud Logica

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“…Since ~ is order-reversing on K, < is the discrete order on K(A, K). Cornish and Fowler (1979) (Theorem 2.2) prove that the Priestley dual of A/Q is the subspace X = {x e. D(y4,2)|x < g(x)} of D(A,2). Hence it remains to prove that =s on K(A, K) is discrete if and only if < on ^ is discrete.…”

Section: (V) P M Jm >N>0) Then For All a B G K K( A B) Is Order-imentioning

confidence: 99%

Lattices of homomorphisms

Davey

Priestley

1986

J Aust Math Soc A

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Given a variety K of lattice-ordered algebras, A e K is catalytic if for all B e K, K(A, B) is a lattice for the pointwise order. The catalytic objects are determined for various varieties of distributivelattice-ordered algebras. The characterisations obtained do not show an overall unity and exhibit diverse behaviour. Duality is employed extensively. Its usefulness in this context depends on the existence of an order-isomorphism between K(A, B) and the corresponding dual hom-set. Criteria for the existence of such an order-isomorphism are investigated for dualities of the Davey-Werner type. The relationship between catalytic objects and colattices is also discussed.

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“…So, we get the second part of (27). (8) and (26) we have (9) and (19)) (6), (13) and (14)) (6), (21) and (14)) [x'x+*(x* + 2*')]* (by (21) and (14)) 0' = 1 (by (14), ( 6 ) , (25) and (12)). U C o r 01 1 a r y 2.3.…”

Section: Preliminariesmentioning

confidence: 94%

“…An algebra ( L , +, ., *, 0 , l ) of type (2,2,1,0,0) is called a pseudocomplemented distributive lattice if ( L , +, ., 0 , 1) is a bounded distributive lattice and *, the unary operation of pseudocomplementation, satisfies the following identities and quasi-identities: (6) 22' = 0, (7) zy = 0 -y 5 2 * , ( 9 ) ( 2 + y)* = z*y*, (10) 2*** = z*,…”

Section: Preliminariesmentioning

confidence: 99%