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Congruences and homomorphisms on Ω -algebras

Elijah Eghosa Edeghagba, Branimir Šešelja, Andreja Tepavčević (2017)

Kybernetika

The topic of the paper are Ω -algebras, where Ω is a complete lattice. In this research we deal with congruences and homomorphisms. An Ω -algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an Ω -valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce Ω -valued congruences, corresponding quotient Ω -algebras and Ω -homomorphisms and we investigate connections among these notions. We prove...

Congruences on pseudocomplemented semilattices

Zuzana Heleyová (2000)

Discussiones Mathematicae - General Algebra and Applications

It is known that congruence lattices of pseudocomplemented semilattices are pseudocomplemented [4]. Many interesting properties of congruences on pseudocomplemented semilattices were described by Sankappanavar in [4], [5], [6]. Except for other results he described congruence distributive pseudocomplemented semilattices [6] and he characterized pseudocomplemented semilattices whose congruence lattices are Stone, i.e. belong to the variety B₁ [5]. In this paper we give a partial solution to a more...

Congruences on semilattices with section antitone involutions

Ivan Chajda (2010)

Discussiones Mathematicae - General Algebra and Applications

We deal with congruences on semilattices with section antitone involution which rise e.g., as implication reducts of Boolean algebras, MV-algebras or basic algebras and which are included among implication algebras, orthoimplication algebras etc. We characterize congruences by their kernels which coincide with semilattice filters satisfying certain natural conditions. We prove that these algebras are congruence distributive and 3-permutable.

Conjugated algebras

Ivan Chajda (2009)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

We generalize the correspondence between basic algebras and lattices with section antitone involutions to a more general case where no lattice properties are assumed. These algebras are called conjugated if this correspondence is one-to-one. We get conditions for the conjugary of such algebras and introduce the induced relation. Necessary and sufficient conditions are given to indicated when the induced relation is a quasiorder which has “nice properties", e.g. the unary operations are antitone...

Convergence in MV-algebras.

George Georgescu, Fortuna Liguori, Giulia Martini (1997)

Mathware and Soft Computing

MV-algebras were introduced in 1958 by Chang [4] and they are models of Lukasiewicz infinite-valued logic. Chang gives a correspondence between the category of linearly ordered MV-algebras and the category of linearly ordered abelian l-groups.Mundici [10] extended this result showing a categorical equivalence between the category of the MV-algebras and the category of the abelian l-groups with strong unit.In this paper, starting from some definitions and results in abelian l-groups, we shall study...

Convex chains in a pseudo MV-algebra

Ján Jakubík (2003)

Czechoslovak Mathematical Journal

For a pseudo M V -algebra 𝒜 we denote by ℓ ( 𝒜 ) the underlying lattice of 𝒜 . In the present paper we investigate the algebraic properties of maximal convex chains in ℓ ( 𝒜 ) containing the element 0. We generalize a result of Dvurečenskij and Pulmannová.

Countable chains of distributive lattices as maximal semilattice quotients of positive cones of dimension groups

Pavel Růžička (2006)

Commentationes Mathematicae Universitatis Carolinae

We construct a countable chain of Boolean semilattices, with all inclusion maps preserving the join and the bounds, whose union cannot be represented as the maximal semilattice quotient of the positive cone of any dimension group. We also construct a similar example with a countable chain of strongly distributive bounded semilattices. This solves a problem of F. Wehrung.

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