Congruences of a Permutable Inverse Semigroup of Finite Rank

Abstract

We describe the structure of any congruence of a permutable inverse semigroup of finite rank.As is known, any two congruences on a group are permutable with respect to the ordinary operation of superposition of binary relations. It is obvious that algebraic structures containing a group structure (rings, moduli, etc.) possess the corresponding property. Equasigroups and finite quasigroups also belong to the class of binary algebras with permutable congruences. As for the theory of semigroups, only classes of semigroups with permutable congruences (Brandt semigroups, all types of finite symmetric semigroups, semigroups of endomorphisms and partial automorphisms of a linearly ordered set, etc.) are known. Necessary and sufficient conditions under which an antigroup of finite rank is permutable (for the definition, see Sec. 1) were established in [1]. In the present paper, we substantially generalize the results of [1]. In Sec. 4 (Theorem 4), we describe the structure of any congruence of a permutable inverse semigroup with zero of finite rank. Main Terminology and NotationA semilattice S is called a semilattice of finite length if there exists a natural number n such that the length of any chain from S does not exceed n.Let P be an ordered set of finite length with smallest element 0. The supremum of the lengths of chains that connect 0 and an element x is called the height of the element x and is denoted by h ( x ).Let S be an arbitrary semigroup and let N 0 be the set of all nonnegative integers. A function rank : S → N 0 is called a rank function on a semigroup S if, for any elements a, b ∈ S, the following inequality is true:The number rank ( a ) is called the rank of the element a.A semigroup is called permutable if any two congruences on it are permutable with respect to the ordinary superposition of binary relations.For other necessary notions from the theory of semigroups and theory of inverse semigroups, see [2, 3]. Rank Function and Its Main PropertiesIn this section, we introduce a rank function on an inverse semigroup whose semilattice of idempotents is of finite length and establish its main properties. To this end, we first define the rank of an element of a semilattice of finite length.

On maximal stable orders on an inverse semigroup of finite rank with zero

“…Therefore, by virtue of Theorem 1 (see [13]), the semigroup S is permutable. We now show that the ideal …”

“…Furthermore, it is obvious that condition 2 of Theorem 1 (see [13]) is satisfied. Therefore, by virtue of Theorem 1 (see [13]), the semigroup S is permutable.…”

“…The function rank group S, is a rank function (see [3], p. 470). We say that an inverse semigroup is an inverse semigroup of finite rank if the semilattice of its idempotents has finite length.…”

“…It is easy to verify that I l = [3]), every ideal of the semigroup S is a rank ideal, and, hence, there exist nonnegative integers k and m such that…”

Structure of a Munn semigroup of finite rank every stable order of which is fundamental or antifundamental

“…Let S be an inverse semigroup whose lattice of idempotents has finite length. The function rank ( ) a = h aa -1 ( ) , where h aa -1 ( ) is the height of the idempotent aa -1 in the semilattice of idempotents of the semigroup S, is a rank function (see [3]). We say that an inverse semigroup is a semigroup of finite rank if the semilattice of its idempotents has finite length.…”