The congruence lattice of a combinatorial strict inverse semigroup

Abstract. Weak relative pseudocomplementation on a meet semilattice S is a partial operation * which associates with every pair (x, y) of elements, where x ≥ y, an element z (the weak pseudocomplement of x relative to y) which is the greatest among elements u such that y = u ∧ x. The element z coincides with the pseudocomplement of x in the upper section [y) and, if S is modular, with the pseudocomplement of x relative to y. A weakly relatively pseudomented semilattice is said to be extended, if it is equipped with a total binary operation extending * . We study congruence properties of the variety of such semilattices and review some of its subvarieties already described in the literature. IntroductionA meet semilattice is said to be weakly relatively pseudocomplemented if, whenever y ≤ x, there is a greatest element u such that y = u ∧ x. The concept goes back to [28], where the congruence lattice of a semilattice was shown to possess this property; the term, however, was introduced later in [35]. Weak relative pseudocomplements in congruence lattices (of various structures) are discussed also, for example, in [2,16,19,41]; they are uncovered as well in lattices of closure operators [18,31], certain subalgebra lattices of semigroups and groups [37,38,42], and in algebraic structures of constraint programming [5,6]. Every meet-semidistributive algebraic (in particular, finite) lattice is weakly relatively pseudocomplemented [12,17,37]. Weak relative pseudocomplementation has been studied also in posets [15]. Sectionally pseudocomplemented semilattices introduced in [8,10] (i.e., semilattices with pseudocomplemented principial filters) are just 1991 Mathematics Subject Classification: 03G25, 06A12, 08A30, 08B12.

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The congruence lattice of a combinatorial strict inverse semigroup

Abstract: The congruence lattice of a combinatorial strict inverse semigroup is shown to be isomorphic to a complete subdirect product of congruence lattices of semilattices preserving pseudocomplements.

Cited by 3 publications

References 9 publications

Semigroups and their lattice of congruences II

Semigroups and their lattice of congruences II

On Incidence Algebras of Combinatorial Inverse Monoids

Weak relative pseudocomplements in semilattices

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