Minimal Paths in Semigroups
by
Zarko Popovic
University of Nis, Faculty of Economics, Trg kralja Aleksandra 11, P. O. Box 121, 18000 Nis, Yugoslavia
Coauthors: Stojan Bogdanovic (University of Nis, Faculty of Economics, Nis, Yugoslavia), Miroslav Ciric (University of Nis, Faculty of Sciences and Mathematics, Nis Yugoslavia)

To an arbitrary semigroup S we can associate the graphs (S, --> ) and (S, —) corresponding respectively to the relations --> and — on S defined as follows:

a --> b <===> ( existsn in N) bn in S1aS1

and

a — b <===> a --> b --> a.

The graph (S, --> ) one considers as a digraph, while (S, —) is an undirected graph.

General properties of these graphs were studied by M. S. Putcha (1974). The well-known theorems of T. Tamura (1972) and M. S. Putcha (1974) assert that two elements a and b belong to the same component in the greatest semillatice decomposition of S if and only if there exists a path from a to b in (S, --> ), or equivalently, if there exists a path between a and b in (S, —). The structure of semigroups in the length of minimal paths in the graph (S, --> ) are bounded was described by S. Bogdanovic and M. Ciric (1996), and the structure of semigroups in which the length of minimal paths in the graph (S, —) are bounded was described by S. Bogdanovic, M. Ciric and Z. Popovic (2000).

If S is a completely \pi-regular semigroup, then the minimal paths from-to, or equivalently, between elements of S in the graphs (S, --> ) and (S, —), which correspond to S, will be described using the minimal paths from-to, or equivalently, between the idempotent elements of S.

Date received: May 30, 2001