Matrix representations of semigroups

D. B. McAlister
1967 Glasgow Mathematical Journal  
In a series of papers [6], [7] , [8] , [10] , Munn has considered the problem of constructing all irreducible representations of a semigroup by matrices over a field. In [10] , he showed how to construct all the irreducible representations of an arbitrary inverse semigroup from those of associated Brandt semigroups. In this paper, we generalize the method of [10] to give a construction for the irreducible representations of an arbitrary semigroup from those of certain associated semigroups. For
more » ... ted semigroups. For many types of semigroups, including regular semigroups, periodic semigroups and 0-simple semigroups with non-zero idempotents, the associated semigroups are completely 0-simple. In this case, by means of Clifford's result [1] on the representations of a completely 0-simple semigroup, we can give an explicit method of construction for all irreducible representations. I should like to express my sincere gratitude to Dr W. D. Munn, who read the first rough draft of these results and who encouraged me to prepare them for publication. 1. ^-semigroups. In general, a semigroup need have neither a zero nor an identity. However, given any semigroup S, we may embed S in a semigroup S° which has a zero and which is constructed from S in the following way. If S already has a zero and contains at least two members, then S=S°; otherwise S° is the semigroup formed from S by adjoining a new symbol 0 and defining a0=0=0a for each a e S° = S u {0}. The phrase " S=S° " means that S is a semigroup which has a zero and at least two members. In a similar way, we can embed a semigroup S in a semigroup S 1 that has an identity. Because of the simple nature of the embedding of a semigroup S in the corresponding semigroup S°, many theorems about semigroups that have no zero may be deduced from corresponding theorems for semigroups that have a zero. In particular, there will be no loss of generality if, in this paper, we consider only semigroups that have a zero. A homomorphism 0 of a semigroup S=S° onto a semigroup 5 is said to be O-restricted if ad=00 implies a=0; the corresponding congruence on S is also said to be O-restricted. PROPOSITION 1. Let S = S° be a semigroup. Then p = {(a, b)eSx S:for alls, teS 1 , sat = 0 if and only ifsbt = 0} is a O-restricted congruence on S. If x is any O-restricted congruence on S, then t £ p. Proof. The relation p is clearly an equivalence on S. Let (a, b)e p,xe S. Then, for any s, teS 1 , sat = 0 if and only if sbt = 0. Hence, a fortiori, saxt = 0 if and only if sbxt = O; thus (ax, bx)ep. Similarly (xa, xb)ep and so p is a congruence on S. Let a e S with (a, 0) e p. Then, for any s, t e S 1 , sat=0; in particular, a=0. Hence (a, 0) e p implies fl=0 so that p is a O-restricted congruence on S. https://doi.
doi:10.1017/s0017089500000033 fatcat:5nayq72ta5cn3ju5a6avmok3rm