Semigroups arising from families of normal subgroups

Given any family of normal subgroups of a group, we construct in a natural 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 of finite boolean type. We also deter!nine when such a monoid is regular, orthodox, or inverse.

Let G be a group and let H = (H-i)(i is an element of M) be a family of normal subgroups of G. Let I-H be the set of 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 composition of mappings, I-H is a monoid. For each alpha is an element of I-H let

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 of 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.