Weakly iterated block products of finite monoids

Abstract

The block product of monoids is a bilateral version of the better known wreath product. Unlike the wreath product, block product is not associative. All decomposition theorems based on iterated block products that have appeared until now have assumed right-to-left bracketing of the operands. We here study what happens when the bracketing is made left-to-right. This parenthesization is in general weaker than the traditional one. We show that weakly iterated block products of semilattices correspond exactly to the well-known variety DA of finite monoids: if groups are allowed as factors, the variety DA*G is obtained. These decomposition theorems allow new, simpler, proofs of recent results concerning the defining power of generalized first-order logic using two variables only.