Global determinism of *-bands

A class of algebras is said to be globally determined if any two members of that class having isomorphic power algebras are isomorphic. In 1984, Y. Kobayashi proved that semilattices are globally determined. By using his methods, we prove that the class of all ∗-bands has the same property. Let 〈A,F〉 be a universal algebra. Clearly, the operations from F can be extended to the collection of all nonempty subsets of A by defining f(A1, . . . , An) = {f(a1, . . . , an) : ai ∈ Ai, 1 ≤ i ≤ n} for all n-ary operations f ∈ F and A1, . . . , An ⊆ A. In the way just described, we obtain the power algebra, or the global of the algebra A, denoted by P(A). In particular, if S is a semigroup, then P(S) is a semigroup too, with the operation given by XY = {xy : x ∈ X, y ∈ Y }, for all nonempty X, Y ⊆ S. A class C of algebras of the same similarity type is said to be globally determined if for each A,B ∈ C the following implication holds: P(A) ∼= P(B) =⇒ A ∼= B. The problem of global determinism was formulated for the first time in the sixties by B. M. Schein, and also in the pioneering paper of Tamura and Shafer [15], where it was proved that the class of all groups possesses this property. Further research in this field was mainly concentrated on various classes of semigroups and related associative systems. Although the class of all semigroups is not globally determined, as proved by Mogiljanskaja [9], a number of relevant globally determined classes of semigroups were found: rectangular groups [12], finite semilattices of torsion groups and finite simple semigroups [5], completely regular periodic monoids (in particular, all bands with an identity element) and their orthogonal sums [6], completely simple and completely 0-simple semigroups [13, 14], and so on. Recall that an involution semigroup is a semigroup equipped with an involutary antiautomorphism. In other words, we are concerned with algebras of 1Presented at the IMC “Filomat 2001”, Nǐs, August 26–30, 2001 2000 Mathematics Subject Classification: 08A99