SEMIGROUPS ARISING FROM FAMILIES OF NORMAL-SUBGROUPS

Given any family of normal subgroups of a group, we construct in a nat ural way a certain monoid, the group of units of which is a semidirect product. We apply this to obtain a description of both the semigroup of endomorphisms and the group of automorphisms of an Ockham algebra o f finite boolean type. We also deter!nine when such a monoid is regula r, orthodox, or inverse. Let G be a group and let H = (H-i)(i is an el ement of M) be a family of normal subgroups of G. Let I-H be the set o f mappings alpha : M --> M such that (For All i is an element of M) H- i subset of or equal to H-alpha(i). It is clear that, under compositio n of mappings, I-H is a monoid. For each alpha is an element of I-H le t S-alpha = X/i is an element of M G/H-alpha(i) x {alpha} and define S (G,H) = boolean OR/alpha is an element of I-H S-alpha. Every element o f S(G,H) is then of the form ((g(i)H(alpha(i)))(i is an element of M), alpha) where alpha is an element of I-H and (g(i))i is an element of M) is a family of elements of G. In what follows, we shall write this in the abbreviated form [g(i)H(alpha(i))]i is an element of M.