On the structure of symmetric n-ary bands

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

On Semilattices of R-Ample Semigroups with Left Central Idempotents

In this article, the semilattice decomposition of r-ample semigroups with left central idempotents is given. By using this decomposition, we show that a semigroup is a r-ample semigroup with left central idempotents if and only if it is a strong semilattice of , where is a monoid and is a right zero band. As a corollary, the characterization theorem of Clifford semigroups is also extended from a strong semilattice of groups to a strong semilattice of right groups. These theories are the basis that the structure theorem of r-ample semigroups with left central idempotents can be established.

Semilattice Decomposition of Locally Associative AG**-groupoids

We consider a locally associative AG **-groupoid S and show that it has associative powers. We define ρ on S as aρb if and only if bna = bn+1 and anb = an+1 for all a, b ∈ S, and show that S/ρ is a maximal separative homomorphic image of S. We show that every S can be uniquely expressible as a semilattice Y of locally associative Archimedean AG **-groupoids Sα(α ∈ Y), the semilattice Y is isomorphic to a maximal homomorphic image S/η of S, and Sα(α ∈ Y) are the equivalence classes of S mod η.

The semigroup of nonempty finite subsets of rationals

LetQbe the additive group of rational numbers and letℛbe the additive semigroup of all nonempty finite subsets ofQ. ForX∈ℛ, defineAXto be the basis of〈X−min(X)〉andBXthe basis of〈max(X)−X〉. In the greatest semilattice decomposition ofℛ, let𝒜(X)denote the archimedean component containingX. In this paper we examine the structure ofℛand determine its greatest semilattice decomposition. In particular, we show that forX,Y∈ℛ,𝒜(X)=𝒜(Y)if and only ifAX=AYandBX=BY. Furthermore, ifXis a non-singleton, then the idempotent-free𝒜(X)is isomorphic to the direct product of a power joined subsemigroup and the groupQ.

The semigroup of nonempty finite subsets of integers

LetZbe the additive group of integers andgthe semigroup consisting of all nonempty finite subsets ofZwith respect to the operation defined byA+B={a+b:a∈A, b∈B}, A,B∈g.ForX∈g, defineAXto be the basis of〈X−min(X)〉andBXthe basis of〈max(X)−X〉. In the greatest semilattice decomposition ofg, letα(X)denote the archimedean component containingXand defineα0(X)={Y∈α(X):min(Y)=0}. In this paper we examine the structure ofgand determine its greatest semilattice decomposition. In particular, we show that forX,Y∈g,α(X)=α(Y)if and only ifAX=AYandBX=BY. Furthermore, ifX∈gis a non-singleton, then the idempotent-freeα(X)is isomorphic to the direct product of the (idempotent-free) power joined subsemigroupα0(X)and the groupZ.

Small varieties of finite semigroups and extensions

We find the atoms of certain subclasses of varieties of finite semigroups and the corresponding varieties of languages. For example we give a new description of languages whose syntactic monoids are R-trivial and idempotent. We also describe the least variety containing all commutative semigroups and at least one non-commutative semigroup. Finally we extend to varieties of finite semigroups some classical results about semilattice decomposition of semigroups.