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Abstract  

In this paper we study the congruences of *-regular semigroups, involution semigroups in which every element is p-related to a projection (an idempotent fixed by the involution). The class of *-regular semigroups was introduced by Drazin in 1979, as the involutorial counterpart of regular semigroups. In the standard approach to *-regular semigroup congruences, one ,starts with idempotents, i.e. with traces and kernels in the underlying regular semigroup, builds congruences of that semigroup, and filters those congruences which preserve the involution. Our approach, however, is more evenhanded with respect to the fundamental operations of *-regular semigroups. We show that idempotents can be replaced by projections when one passes from regular to *-regular semigroup congruences. Following the trace-kernel balanced view of Pastijn and Petrich, we prove that an appropriate equivalence on the set of projections (the *-trace) and the set of all elements equivalent to projections (the *-kernel) fully suffice to reconstruct an (involution-preserving) congruence of a *-regular semigroup. Also, we obtain some conclusions about the lattice of congruences of a *-regular semigroup.

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Abstract

The aim of this paper is to study the congruences on abundant semigroups with quasi-ideal adequate transversals. The good congruences on an abundant semigroup with a quasi-ideal adequate transversal S° are described by the equivalence triple abstractly which consists of equivalences on the structure component parts I, S° and Λ. Also, it is shown that the set of all good congruences on this kind of semigroup forms a complete lattice.

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Abstract  

We prove that the tolerance lattice TolA of an algebra A from a congruence modular variety V is 0-1 modular and satisfies the general disjointness property. If V is congruence distributive, then the lattice Tol A is pseudocomplemented. If V admits a majority term, then Tol A is 0-modular.

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modularity for congruence lattices of algebras , Canad. Math. Bull., 12 ( 1969 ), 167 - 173 . [2] Gumm , H.-P. , Geometrical methods in congruence modular algebras , Mem,. Amer. Math. Soc., 45 ( 1983 ). [3] Hobby , D. and McKenzie , R. N

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. [3] E dwards , P. M. , Maximizing a congruence with respect to its partition of idempotents , Semigroup Forum , 39 ( 1985 ), 313 – 317 . [4

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We survey results concerning the representations of lattices as lattices of congruences and as lattices of equational theories. Recent results and open problems will be mentioned.

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Abstract  

We show that the polynomials of every finite Mal’cev algebra with congruence lattice of height at most 2 can be described by a finite set of relations.

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Abstract  

The relationship between absolute retracts, injectives and equationally compact algebras in finitely generated congruence distributive varieties with 1- element subalgebras is considered and several characterization theorems are proven. Amongst others, we prove that the absolute retracts in such a variety are precisely the injectives in the amalgamation class and that every equationally compact reduced power of a finite absolute retract is an absolute retract. We also show that any elementary amalgamation class is Horn if and only if it is closed under finite direct products.

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Alimpić, B. P. and Krgović, D. N. , Some congruences on regular semigroups, Lect. Notes Math. , 1320 , Springer-Verlag (1986), 1–10. Krgović D. N

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