Rees matrix seminearring

Abstract

As a continuation of the work done in (R. Mukherjee (Pal), P. Pal and S. K. Sardar, On additively completely regular seminearrings, Commun. Algebra  45 (12) (2017) 5111–5122), in this paper, our objective is to characterize left (right) completely simple seminearrings in terms of Rees Construction by generalizing the concept of Rees matrix semigroup (J. M. Howie, Fundamentals of Semigroup Theory (Clarendon Press, Oxford, 1995); M. Petrich and N. R. Reilly, Completely Regular Semigroups (Wiley, New York, 1999)) and that of Rees matrix semiring (M. K. Sen, S. K. Maity and H. J. Weinert, Completely simple semirings, Bull. Calcutta Math. Soc.  97 (2005) 163–172). In Rees theorem, a completely simple semigroup is coordinatized in such a way that each element can be seen to be a triplet which gives this abstract structure a much more simpler look. In this paper, we have been able to construct a similar kind of coordinate structure of a restricted class of left (right) completely simple seminearrings taking impetus from (M. P. Grillet, Semirings with a completely simple additive semigroup, J. Austral. Math. Soc.  20 (Ser. A) (1975) 257–267, Theorem [Formula: see text] and (M. K. Sen, S. K. Maity and H. J. Weinert, Completely simple semirings, Bull. Calcutta Math. Soc.  97 (2005) 163–172, Theorem 3.1).