Regularity Equations and Conditions on Semigroups
by
Miroslav Ciric
University of Nis, Nis, Yugoslavia
Coauthors: Stojan Bogdanovic (University of Nis), Predrag Stanimirovic (University of Nis), Tatjana Petkovic (University of Nis and TUCS, Turku, Finland)

Let W be the set of all words u over a countable alphabet X \cup {c} with c not in X, such that u =/= c, c appears at least once in u and any letter from X appears at most once in u. The letters from X are called variables , c is called a constant , we write u(c, x₁, ... , x_n) instead of u to emphasize that {x₁, ... , x_n} is the set of all letters from X which appear in u, and for a semigroup S and a, a₁, ... , a_n in S, u(a, a₁, ... , a_n) denotes the element of S obtained by replacement of c by a, x_i by a_i and the concatenation in the free semigroup (X \cup {c})⁺ by the multiplication in S.

An expression c=u(c, x₁, ... , x_n) we call a (linear ) equation , if S is a semigroup and a in S, an expression a=u(a, x₁, ... , x_n) we call an equation in S, which is said to be solvable in S if there exist a₁, ... , a_n in S such that a=u(a, a₁, ... , a_n) and then a is said to be a u-regular element . As known, if a=axa, a=xa², a=a²x, a=a²xa², a=xa²y, c=xcyc, c=cxcy, c=xcycz, c=xc, c=cx or c=xcy is solvable, then a is called a regular, left regular, right regular, completely regular, intra-regular, left quasi-regular, right quasi-regular, semisimple, left reproduced, right reproduced or reproduced element, respectively. On the other hand, an expression <c=u(c, x₁, ... , x_n)> we call a regularity condition and we say that a semigroup S satisfies this condition, or that it is a u-regular semigroup , if every its element is u-regular.

The main purpose of our investigations is to prove that there are infinitely many ways to define regularity of elements by (linear) equations, but that all classes of regularity can be classified into 16 types, and we describe the partially ordered set of classes of regularity. We also prove that every regularity condition is equivalent to one of 14 conditions and describe the partially ordered set of classes of regularity conditions.

Date received: May 12, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-30.