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Free trees and the optimal bound in Wehrung's theorem

Pavel Růžička (2008)

Fundamenta Mathematicae

We prove that there is a distributive (∨,0,1)-semilattice of size ℵ₂ such that there is no weakly distributive (∨,0)-homomorphism from C o n c A to with 1 in its range, for any algebra A with either a congruence-compatible structure of a (∨,1)-semi-lattice or a congruence-compatible structure of a lattice. In particular, is not isomorphic to the (∨,0)-semilattice of compact congruences of any lattice. This improves Wehrung’s solution of Dilworth’s Congruence Lattice Problem, by giving the best cardinality...

Generalized deductive systems in subregular varieties

Ivan Chajda (2003)

Mathematica Bohemica

An algebra 𝒜 = ( A , F ) is subregular alias regular with respect to a unary term function g if for each Θ , Φ ∈ Con 𝒜 we have Θ = Φ whenever [ g ( a ) ] Θ = [ g ( a ) ] Φ for each a ∈ A . We borrow the concept of a deductive system from logic to modify it for subregular algebras. Using it we show that a subset C ⊆ A is a class of some congruence on Θ containing g ( a ) if and only if C is this generalized deductive system. This method is efficient (needs a finite number of steps).

Green's relations and their generalizations on semigroups

Kar-Ping Shum, Lan Du, Yuqi Guo (2010)

Discussiones Mathematicae - General Algebra and Applications

Green's relations and their generalizations on semigroups are useful in studying regular semigroups and their generalizations. In this paper, we first give a brief survey of this topic. We then give some examples to illustrate some special properties of generalized Green's relations which are related to completely regular semigroups and abundant semigroups.

Ideal extensions of graph algebras

Karla Čipková (2006)

Czechoslovak Mathematical Journal

Let 𝒜 and ℬ be graph algebras. In this paper we present the notion of an ideal in a graph algebra and prove that an ideal extension of 𝒜 by ℬ always exists. We describe (up to isomorphism) all such extensions.

Implication algebras

Ivan Chajda (2006)

Discussiones Mathematicae - General Algebra and Applications

We introduce the concepts of pre-implication algebra and implication algebra based on orthosemilattices which generalize the concepts of implication algebra, orthoimplication algebra defined by J.C. Abbott [2] and orthomodular implication algebra introduced by the author with his collaborators. For our algebras we get new axiom systems compatible with that of an implication algebra. This unified approach enables us to compare the mentioned algebras and apply a unified treatment of congruence properties....

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