Semigroups whose right ideals are finitely generated

We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterize weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterize weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal {H}$ is finitely generated.

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.

Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces

This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known that braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.

The strong nil-cleanness of semigroup rings

In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M={ {\mathcal M} }^{0}(G;I,\text{Λ};P) , we show that the contracted semigroup ring {R}_{0}{[}M] is strongly nil-clean if and only if either |I|=1 or |\text{Λ}|=1 , and R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R{[}S] is strongly nil-clean if and only if R{[}{S}_{\alpha }] is strongly nil-clean for each \alpha \in Y .

The Strong Semilattice of Pi-groups

A semigroup is called a GV-inverse semigroup if and only if it is isomorphic to a semilattice of $\pi$-groups. In this paper, we give the sufficient and necessary conditions for a GV-inverse semigroup to be a strong semilattice of $\pi$-groups. Some conclusions about Clifford semigroups are generalized.

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.

GREEN ♯-RELATIONS AND NORMAL ${\cal H}^\sharp$-CRYPTOGROUPS

In this paper, a new set of generalized Green's relations on a semigroup S, namely the set of Green ♯-relations, are introduced. By using these new Green ♯-relations, we prove that an [Formula: see text]-abundant semigroup is a normal [Formula: see text]-cryptogroup if and only if it is a strong semilattice of completely [Formula: see text]-simple cryptogroups. Our theorem simplifies a construction theorem of normal [Formula: see text]-abundant cryptographs previously described by the authors. Some properties of the good homomorphisms between the normal [Formula: see text]-cryptogroups are also investigated.

A Structure Theorem of Regular ${\cal H}^\sharp$-Cryptographs

A new set of generalized Green relations is given in studying the [Formula: see text]-abundant semigroups. By using the generalized strong semilattice of semigroups recently developed by the authors, we show that an [Formula: see text]-abundant semigroup is a regular [Formula: see text]-cryptograph if and only if it is an [Formula: see text]-strong semilattice of completely [Formula: see text]-simple semigroups. This result not only extends the well known result of Petrich and Reilly from the class of completely regular semigroups to the class of semiabundant semigroups, but also generalizes a well known result of Fountain on superabundant semigroups from the class of abundant semigroups to the class of semiabundant semigroups.