Regularity of Semihypergroups of Infinite Matrices

S. Chaopraknoi, N. Triphop

Abstract

A semigroup S is a regular semigroup if for every $x \in S$, x = xyx for some $y \in S$, and a semihypergroup $(H,\circ)$ is a regular semihypergroup if for every $x \in H$, $x \in x \circ y \circ x $ for some $y \in H$. If S is a semigroup and P is a nonempty subset of S, we let (S, P) denote the semihypergroup (S,\circ) where $x \circ y = xPy$ for all $x, y \in S$. Let BM(F) be the multiplicative semigroup of all bounded $N \times N$ matrices over a field F where N is the set of natural numbers. It is known that BM(F) is a regular semigroup. Our purpose is to provide necessary and sufficient conditions for a nonempty subset P of BM(F) so that (BM(F);P) is a regular semihypergroup.