Abstract. A semigroup $S$ with zero is {\it categorical at zero} or a {\it C-semigroup} if given any $a,b,c\in S$, where $ab,bc\in S\setminus\{0\} $, then it follows that $abc\in S\setminus\{0\} $. A semigroup with zero is {\it left (right) idempotent bounded} if it is the $0$-direct union of idempotent generated principal left (right) ideals and {\it idempotent bounded} if it is both left and right idempotent bounded. We abbreviate these concepts by LIB, RIB and IB respectively. The aim of this paper is to give a structure theorem for the class of LIB C-semigroups in terms of what we call {\it left blocked Rees matrix semigroups}. In a sequel, we specialize our work to describe IB C-semigroups in terms of {\it double blocked Rees matrix semigroups}. The classes of LIB C-semigroups and IB C-semigroups are extremely large, including the classes of primitive abundant semigroups and completely $0$-simple semigroups. Consideration of semigroups that are idempotent bounded on the left only, allows us to move away from structure theorems for C-semigroups satisfying conditions that are inherently two sided, such as regularity.

AMS Subject Classification (1991): 20M10

Keyword(s): left blocked Rees matrix semigroups

Received October 10,1991, and in revised form November 20,1992. (Registered under 5531/2009.)