Square extension of a groupoid
by
Adam Marczak
Wroclaw University of Technology
Coauthors: Jerzy Plonka (Polish Academy of Sciences, Mathematical Institute)

The problem of the clone characterization of algebras posed in late sixties by Marczewski exerted a great influence on the modern algebra. So called p_n-sequences of algebras introduced by Grätzer in 1969 proved to be not only a good description of the complexity of algebras but also, to some extent, they can characterize algebras.

A characterization of algebras with exactly n different essentially n-ary term operations was given by Ponka in 1971. Then, a natural question about the possibilities of an extension of this result into another classes of algebras appeared. An equational characterization of all semigroups (and, in fact, all groupoids) having exactly n+1 different essentially n-ary term operations was presented by Marczak during AAA59, Potsdam, February 2000. It was proved that if a semigroup without algebraic constants has exactly n+1 different essentially n-ary term operations for every positive integer n, then it belongs to one of two different varieties of groupoids. Moreover, the sufficient conditions for algebras from these varieties to represent the sequence (0, 2, 3,  ... , n,  n+1,  ...) were formulated.

This talk origins from the investigation of a structural characterization of algebras from the varieties found by Marczak. A new construction called a square extension of a groupoid is here introduced. We present properties of this construction and give some examples. In particular, a modification of the construction called the associative square extension of a band is applied to a structural characterization of semigroups with exactly n+1 different essentially n-ary term operations for every positive integer n.

Let I =(I;  o ) be an idempotent groupoid and let {A_i}_{i in I} be a family of pairwise disjoint sets. Assume that the family {A_i}_{i in I} satisfies the following conditions:

i in A_i for every i in I,

a mapping h_ij:A_i --> A_{i o j} is given for every i, j in I,

h_ij(i)=i o j for every i, j in I,

h_ii(a)=i for every a in A_i.

For the groupoid I we construct a new algebra A_I =(A; ·), where A= \cup _{i in I} A_i and the binary operation · is given by the formula

if a in A_i and b in A_j we put a·b=h_ij(a).

The groupoid A_I we shall call the square extension of the groupoid I. If additionally I is a band and A_I satisfies

h_{i o j, k}(h_ij(x))=h_{i, j o k}(x),

Date received: May 20, 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 # caee-45.