Two-sorted Algebras Research Articles

A duality for two-sorted lattices

A series of representation theorems (some of which discovered very recently) present an alternative view of many classes of algebras related to non-classical logics (e.g. bilattices, semi-De Morgan, Nelson and quasi-Nelson algebras) as two-sorted algebras in the sense of many-sorted universal algebra. In all the above-mentioned examples, we are in fact dealing with a pair of lattices related by two meet-preserving maps. We use this insight to develop a Priestley-style duality for such structures, mainly building on the duality for meet-semilattices of G. Bezhanishvili and R. Jansana. Our approach simplifies all the existing dualities for these algebras and is applicable more generally; in particular, we show how it specialises to the class of quasi-Nelson algebras, which has not yet been studied from a duality point of view.

Dynamic Łukasiewicz Logic and Dynamic MV-algebras

Following K. Segerberg [22], D. Kozen [15] and V. Pratt [19], who have been introduced dynamic propositional logic and dynamic algebras, dynamic propositional Łukasiewicz logic DPŁ (dynamic n-valued propositional Łukasiewicz logic DPŁn) and dynamic MV-algebras (dynamic MVn-algebras) are introduced and theories of the logic DPŁ (DPŁn) and dynamic MV-algebras (MVn-algebras) are developed. Dynamic MV-algebras (dynamic MVn-algebras) are algebraic counterparts of the logic DPŁ (DPŁn), that in turn represent two-sorted algebras that combine the varieties of MV-algebras (MVn-algebras) (M,⊕,⊙,∼,0,1) and regular algebras (R,∪,;,⁎) into a single finitely axiomatized variety (M,R,◇) resembling R-module with “scalar” multiplication ◇. Kripke semantics is developed for dynamic propositional Łukasiewicz logic (dynamic n-valued propositional Łukasiewicz logic DPŁn).

Algebraic properties of if-then-else and commutative three-valued tests

This paper establishes a finite axiomatization of possibly non-halting computer programs and tests, with the if-then-else operation. The model is a two-sorted algebra, with one sort being the programs and the other being the tests. The main operation on programs is composition, and 1 and 0 represent the programs skip and loop (i.e. never halts) respectively. Programs are modeled as partial functions on some state space [Formula: see text], with tests modeled as partial predicates on [Formula: see text]. The operations on the tests are the usual logical connectives ∧, ∨, [Formula: see text], [Formula: see text] and [Formula: see text]. In addition, there is the hybrid operation of if-then-else, and the test-valued operation [Formula: see text] on programs which is true when a program halts, and undefined otherwise. The halting operation [Formula: see text] implies that operations of domain [Formula: see text] and domain join ∨ may also be expressed. When tests are assumed to be possibly non-halting, the evaluation strategy of the logical connectives affects the result. Here we model parallel evaluation, as opposed to the common sequential (or short-circuit) evaluation strategy. For example, we view [Formula: see text] as false if either [Formula: see text] or [Formula: see text] is false, even if the other does not halt.

Conformal algebras, vertex algebras, and the logic of locality

AbstractIn a new algebraic approach, conformal algebras and vertex algebras are extended to two-sorted structures, with an additional component encoding the logical properties of locality. Within these algebras, locality is expressed as an identity, without the need for existential quantifiers. Two-sorted conformal algebras form a variety of two-sorted algebras, an equationally-defined class, and free conformal algebras are given by standard universal algebraic constructions. The variety of two-sorted conformal algebras is equivalent to a Mal’tsev variety of single-sorted algebras. Motivated by a question of Griess, subalgebras of reducts of conformal algebras are shown to satisfy a set of quasi-identities. The class of two-sorted vertex algebras does not form a variety, so open problems concerning the nature of that class are posed.

Fuzzy sets as two-sorted algebras

This paper makes the attempt to explain the foundations of fuzzy sets from the point of view of universal algebra. The free fuzzy set and the fuzzy power set are constructed. Moreover, fuzzy power sets give rise to a monad. In this context, fuzzy power sets appear as free fuzzy modules generated by their underlying fuzzy sets.

Applicative theories for logarithmic complexity classes

We present applicative theories of words corresponding to weak, and especially logarithmic, complexity classes. The theories for the logarithmic hierarchy and alternating logarithmic time formalise function algebras with concatenation recursion as main principle. We present two theories for logarithmic space where the first formalises a new two-sorted algebra which is very similar to Cook and Bellantoni's famous two-sorted algebra B for polynomial time [4]. The second theory describes logarithmic space by formalising concatenation- and sharply bounded recursion. All theories contain the predicates W representing words, and V representing temporary inaccessible words. They are inspired by Cantini's theories [6] formalising B.

Fusion of sequent modal logic systems labelled with truth values

Fusion is a well-known form of combining normal modal logics endowed with a Hilbert calculi and a Kripke semantics. Herein, fusion is studied over logic systems using sequent calculi labelled with truth values and with a semantics based on a two-sorted algebra allowing, in particular, the representation of general Kripke structures. A wide variety of logics, including non-classical logics like, for instance, modal logics and intuitionistic logic can be presented by logic systems of this kind. A categorical approach of fusion is defined in the context of these logic systems. Preservation of soundness and completeness by fusion is studied. Soundness is preserved without further requirements, completeness is preserved under mild assumptions.

Which two-sorted algebras of booleans and naturals have a finite basis?

We show that the two-sorted algebra of Booleans and naturals with conjunction, addition and inequality is not finitely based. If addition is removed, or negation is included, then the resulting algebra is finitely based.

Axiomatizing the subsumption and subword preorders on finite and infinite partial words

We consider two-sorted algebras of finite and infinite partial words equipped with the subsumption preorder and the operations of series and parallel product and omega power. It is shown that the valid equations and inequations of these algebras can be described by an infinite collection of simple axioms, and that no finite axiomatization exists. We also prove similar results for two related preorders, namely for the induced partial subword preorder and the partial subword preorder. Along the way of proving these results, we provide a concrete description of the free algebras in the corresponding varieties in terms of generalized series–parallel partial words.

Peirce algebras

Abstract We present a two-sorted algebra, called a Peirce algebra , of relations and sets interacting with each other. In a Peirce algebra, sets can combine with each other as in a Boolean algebra, relations can combine with each other as in a relation algebra, and in addition we have both a set-forming operator on relations (the Peirce product of Boolean modules) and a relation-forming operator on sets (a cylindrification operation). Two applications of Peirce algebras are given. The first points out that Peirce algebras provide a natural algebraic framework for modelling certain programming constructs. The second shows that the so-called terminological logics arising in knowledge representation have evolved a semantics best described as a calculus of relations interacting with sets.