Semigroups whose lattice of congruences is Boolean
Howard B. Hamilton, THOMAS E. NORDAHL · Pacific Journal of Mathematics · 1978
The commutative semigroups whose lattice of congruences forms a Boolean lattice are determined.They are (i) the null semigroups of order two or less, (ii.) the discrete trees, (iii) the groups which are a direct sum of prime order cyclic groups in which no two factors have the same order, (iv) the semigroups which are a one element inflation of a discrete tree, (v) the semigroups which are a free product of a discrete tree with zero and a semigroup of type (iii) amalgamated over the trivial semigroup, and (vi) the semigroups which are a one element inflation of a semigroup of type1* Introduction* In this paper a semilattice will be considered to be an upper semilattice (i.e., for x,yeS we have x <> y if and only if xy = y).We say that semilattice S is a (discrete) tree if for all x, y eS with x<>y the interval [x, y] -{z eS: x ^ z <> y) is a (finite) chain.For any semigroup S we will let L(S) denote the lattice of congruences of S. If T is a subsemigroup of semigroup S and there exists a function /: S -> Γ satisfying: (i) the restriction of / to T is the identity mapping and (ii) for x, y eS we have xyf(x)f(y), then S is said to be an inflation of T. We shall say that S is a one element inflation of T if S is an inflation of T and [S\T] -1.Terms which are not defined may be found in [3], [9], [12], [13] or in [1].We now present some results which are needed for our proofs. THEOREM 1. (Hamilton [7]) Let S be a semilattice. L(S) is a Boolean lattice if and only if S is a discrete tree.Let S be a semigroup and let S = \JaerS a be the greatest semilattice decomposition of S. If a < β in Γ and for all aeS a and b e S β we have ab = ba -b then we say that S a and S β are 1-composed.If S a and S β are 1-composed for all a, β e Γ with a < β then we say that S is 1-composed.THEOREM 2. (Hamilton [8]) If S is a commutative seperative semigroup with L(S) a modular lattice then S is 1-composed.If a semigroup S is isomorphic to a subdirect product of semigroups T and U we shall write: S ~ T X S U.