1,721,153 research outputs found
Residuated relational systems
No full textThe aim of the present paper is to generalize the concept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the definition of adjointness in such a way that the ordering relation can be harmlessly replaced by a binary relation. By enriching such binary relation with additional properties, we get interesting properties of residuated relational systems which are analogical to those of residuated posets and lattices
A Note on Orthomodular Lattices
No full textWe introduce a new identity equivalent to the orthomodular law in every ortholattice
On residuation in paraorthomodular lattices
No full textParaorthomodular lattices are quantum structures of prominent importance within the framework of the logico-algebraic approach to (unsharp) quantum theory. However, at the present time it is not clear whether the above algebras may be regarded as the algebraic semantic of a logic in its own right. In this paper, we start the investigation of material implications in paraorthomodular lattices by showing that any bounded modular lattice with antitone involution A can be converted into a left-residuated groupoid if it satisfies a strengthened form of regularity. Moreover, the above condition turns out to be also necessary whenever A is distributive
On residuation in paraorthomodular lattices
No full textParaorthomodular lattices are quantum structures of prominent importance within the framework of the logico-algebraic approach to (unsharp) quantum theory. However, at the present time it is not clear whether the above algebras may be regarded as the algebraic semantic of a logic in its own right. In this paper, we start the investigation of material implications in paraorthomodular lattices by showing that any bounded modular lattice with antitone involution A can be converted into a left-residuated groupoid if it satisfies a strengthened form of regularity. Moreover, the above condition turns out to be also necessary whenever A is distributive
Compatible idempotent terms in universal algebra
No full textIn universal algebra, we oftentimes encounter varieties that are not especially well-behaved from any point of view, but are such that all their members have a well-behaved core, i.e. subalgebras or quotients with satisfactory properties. Of special interest is the case in which this coreis a retract determined by an idempotent endomorphism that is uniformly term denable (through a unary term t(x)) in every member of the given variety. Here, we try to give a unied account of this phenomenon. In particular, we investigate what happens when various congruence properties like congruence distributivity, congruence permutability or congruence modularity are not supposed to hold unrestrictedly in any A 2 V, but only for congruence classes of values of the term operation tA
Representing quantum structures as near semirings
No full textIn this article, we introduce the notion of near semiring with involution. Generalizing the theory of semirings we aim at represent quantum structures, such as basic algebras and orthomodular lattices, in terms of near semirings with involution. In particular, after discussing several properties of near semirings, we introduce the so-called Łukasiewicz near semirings, as a particular case of near semirings, and we show that every basic algebra is representable as (precisely, it is term equivalent to) a near semiring. In the particular case in which a Łukasiewicz near semiring is also a semiring, we obtain as a corollary a representation of MV-algebras as semirings. Analogously, by introducing a particular subclass of Łukasiewicz near semirings, that we termed orthomodular near semirings, we obtain a representation of orthomodular lattices. In the second part of the article, we discuss several universal algebraic properties of Łukasiewicz near semirings and we show that the variety of involutive integral near semirings is a Church variety. This yields a neat equational characterization of central elements of this variety. As a byproduct of such, we obtain several direct decomposition theorems for this class of algebras
A semiring-like representation of lattice pseudoeffect algebras
No full textIn order to represent lattice pseudoeffect algebras, a non-commutative generalization of lattice effect algebras, in terms of a particular subclass of near semirings, we introduce in this article the notion of near pseudoeffect semiring. Taking advantage of this characterization, in the second part of the present work, we present, as an application, an alternative, rather straight as well as simple, explanation of the relationship between lattice pseudoeffect algebras and pseudo-MV algebras by means of a simplified axiomatization of generalized Łukasiewicz semirings, a variety of non-commutative semirings equipped with two antitone unary operations
Tolerances as images of congruences in varieties defined by linear identities
No full textAn identity s = t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra A in a variety defined by a set of linear identities. We prove that there exist an algebra B in the same variety and a congruence theta of B such that a homomorphism from B onto A maps theta onto T
A semiring-like representation of lattice pseudoeffect algebras
No full textIn order to represent lattice pseudoeffect algebras, a non-commutative generalization of lattice effect algebras, in terms of a particular subclass of near semirings, we introduce in this article the notion of near pseudoeffect semiring. Taking advantage of this characterization, in the second part of the present work, we present, as an application, an alternative, rather straight as well as simple, explanation of the relationship between lattice pseudoeffect algebras and pseudo-MV algebras by means of a simplified axiomatization of generalized ukasiewicz semirings, a variety of non-commutative semirings equipped with two antitone unary operations
On some properties of directoids
No full textWe study some properties of directoids and their expansions by additional signature, including bounded involutive directoids and complemented directoids. Among other results, we provide a shorter proof of the direct decomposition theorem for bounded involutive directoids given in Chajda and Langer (Directoids. An algebraic approach to ordered sets. Heldermann Verlag, Lemgo 2011); we present a description of central elements of complemented directoids; we show that the variety of directoids, as well as its expansions mentioned above, all have the strong amalgamation property
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