In this paper, for each positive integer m, we associate with a finite monoid S0 and m finite commutative monoids S1,..., Sm, a product ◊m(Sm,..., S1, S0) . We give a representation of the free objects in the pseudovariety ◊m(Wm,..., W1, W0) generated by these (m + 1)-ary products where Si ∈ Wi for all 0 ≤ i ≤ m. We then give, in particular, a criterion to determine when an identity holds in ◊m...

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to ...

An inverse semigroup is a semigroup S such that for each x e S there exists a unique inverse x~ & S such that both xx~x=x and x~xx~=x~\ This condition is equivalent to S both being (von Neumann) regular and having all idempotents commute. The classic example of such a semigroup is the symmetric inverse semigroup Ix of all partial bijections on a set X under the standard composition of partial f...

i.e., each principal filter is pseudo-complemented. Based on the way relatively complemented lattices are defined (every interval is complemented), this seemed like the natural way to define relatively pseudocomplemented (for a lattice with a greatest element, (1) is equivalent to each each interval being pseudo-complemented). However, the term relatively pseudo-complemented was already in use ...

We study the class of symmetric [Formula: see text]-ary bands. These are semigroups text] such that is invariant under action permutations and idempotent, i.e., satisfies for all text]. first provide a structure theorem these bands extends classical (strong) semilattice decomposition certain classes introduce concept strong we show exactly semilattices extensions Abelian groups whose exponents ...