Acts over Right, Left Regular Bands and Semilattices Types

Let S be a semigroup and let X be a non-empty set. Then X is called a right act over S or simply S-act if there is a mapping X x S 4 X , (2 , s) C) x s with the property ( x s ) t = x ( s t ) . A semigroup S is called a band if every element in S is an idempotent. A band S is called r ight regular (resp. l e f t regular) if sts = st (resp. sts = ts) holds for every s, t E S. A commutative band is called a semilatt ice. An S-act X is said to be a right regular band type, or simply RRB-type, if xs2 = x s and x s t s = x s t for all x E X and every s, t E S. A left regular band type (LRB-type) S-act and a semilattice type (SL-type) S-act are similarly defined. When S is a free monoid a RRB-type automaton, an LRB-type automaton and an SL-type auromaton can be similarly defined. In this case, for an automaton A = (A, X , a), where A is an alphabet, X is a set of states and 6 is a mapping X x A -+ X , ( 2 , a) C) xu. we can show that, if xu2 = za and xaba = xab for all x E X , a , b E A, then xs2 = x s and x s t s = x for all x E X, s, t E A*. This fact can be applied to LRB-type automata and SL-type automata. Our purpose is to determine all S-act which are right regular band types, left regular band types and semilattice types, respectively. To achieve the purpose, we obtain necessary and sufficient conditions, for any given set X , and any semigroup S, in order that X is S-acts which are a RRB-type, a LRB-type and a SL-type, respectively (Theorems 1,3,5). Further we obtain more concrete results to construct actually RRB-type, LRB-type and SGtype automata, respectively (Corollaries 2,4,5). Let X be a S-act. It is well-known that defining a relation p on S by spt if x s = x t for all x E X . a transformation semigroup S/p on X can be obtained. Thus, from the above results, every right regular band, left regular band and semilattce can be obtained in the full transformation semigroup T ( X ) , respectively.