Permutative Semigroups Whose Congruences Form a Chain ∗

Semigroups whose congruences form a chain are often termed ∆-semigroups. The commutative ∆-semigroups were determined by Schein and by Tamura. A natural generalization of commutativity is permutativity: a semigroup is permutative if it satisfies a non-identity permutational identity. We completely determine the permutative ∆-semigroups. It turns out that there are only six noncommutative examples, each of which has at most three elements. A semigroup is called permutative if it satisfies an identity x1x2 . . . xn = xσ(1)xσ(2) · · ·xσ(n) , for some non-identity permutation σ of {1, 2, . . . , n} . A ∆-semigroup is one whose congruences form a chain. The commutative ∆-semigroups were completely determined by B. Schein [12], [13] and T. Tamura [15]. In conjunction with their result, stated below as Result 1, our main theorem completely determines the permutative ∆-semigroups: Theorem 1. A semigroup S is a permutative ∆-semigroup if and only if it satisfies one of the following conditions. (i) S is a commutative ∆-semigroup. (ii) S is isomorphic to either R or R , where R is a two-element right zero semigroup. (iii) S is isomorphic to the semigroup Z = {0, e, a} , obtained by adjoining to a null semigroup {0, a} an idempotent element e that is both a right identity and a left annihilator for Z . (iv) S is isomorphic to the dual of a semigroup of type (ii) or (iii). Let R denote the semigroup of positive real numbers under addition and let Q denote the Rees quotient semigroup by the ideal I = [1,∞). Similarly, let R denote the Rees quotient semigroup by the ideal I = (1,∞). A subsemigroup G of Q or R is 0-unitary if x, x+ y ∈ G, x+ y ∈ I together imply y ∈ G . ∗ The first author’s research was supported by the Hungarian NFSR Grant No. T042481 and No. T043034.