On a Product of Finite Monoids

Abstract

In this paper, for each positive integer m, we associate with a finite monoid S0 and m finite commutative monoids S 1 ,…, S m , a product &#x25CA m (S m ,…, S 1 , S 0 ). We give a representation of the free objects in the pseudovariety &#x25CA m ( W m ,…, W 1 , W 0 ) generated by these ( m + 1)-ary products where S i &#x2208 W i for all 0 &#x2264 i &#x2264 m . We then give, in particular, a criterion to determine when an identity holds in &#x25CA m ( J 1 ,…, J 1 , J 1 ) with the help of a version of the Ehrenfeucht-Fraisse game ( J 1 denotes the pseudovariety of all semilattice monoids). The union &#x222A m >0 &#x25CA m ( J 1 ,…, J 1 , J 1 ) turns out to be the second level of the Straubing’s dot-depth hierarchy of aperiodic monoids.