On orthodox P-restriction semigroups

The investigation of orthodox [Formula: see text]-restriction semigroups was initiated by Jones in 2014 as generalizations of orthodox [Formula: see text]-semigroups. The aim of this paper is to further study orthodox [Formula: see text]-restriction semigroups based on the known results of Jones. After establishing a construction theorem for orthodox [Formula: see text]-restriction semigroups, we introduce proper [Formula: see text]-restriction semigroups (which are necessarily orthodox) and prove that every (finite) orthodox [Formula: see text]-restriction semigroup has a (finite) proper cover. Our results enrich and extend existing results for restriction semigroups and orthodox [Formula: see text]-semigroups.

Congruences on glrac semigroups (I)

The theory of congruences on semigroups is an important part in the theory of semigroups. The aim of this paper is to study [Formula: see text]-congruences on a glrac semigroup. It is proved that the [Formula: see text]-congruences on a glrac semigroup become a complete sublattice of its lattice of congruences. Especially, the structures of left restriction semigroup [Formula: see text]-congruences and the projection-separating [Formula: see text]-congruences on a glrac semigroup are established. Also, we demonstrate that they are both complete sublattice of [Formula: see text]-congruences and consider their relations with respect to complete lattice homomorphism.

Some algebraic structures on the generalization general products of monoids and semigroups

For arbitrary monoids A and B, in Cevik et al. (Hacet J Math Stat 2019:1–11, 2019), it has been recently defined an extended version of the general product under the name of a higher version of Zappa products for monoids (or generalized general product) $$A^{\oplus B}$$ A ⊕ B $$_{\delta }\bowtie _{\psi }B^{\oplus A}$$ δ ⋈ ψ B ⊕ A and has been introduced an implicit presentation as well as some theories in terms of finite and infinite cases for this product. The goals of this paper are to present some algebraic structures such as regularity, inverse property, Green’s relations over this new generalization, and to investigate some other properties and the product obtained by a left restriction semigroup and a semilattice.

On E-semiabundant semigroups with a multiplicative restriction transversal

Multiplicative inverse transversals of regular semigroups were introduced by Blyth and McFadden in 1982. Since then, regular semigroups with an inverse transversal and their generalizations, such as regular semigroups with an orthodox transversal and abundant semigroups with an ample transversal, are investigated extensively in literature. On the other hand, restriction semigroups are generalizations of inverse semigroups in the class of non-regular semigroups. In this paper we initiate the investigations of E-semiabundant semigroups by using the ideal of "transversals". More precisely, we first introduce multiplicative restriction transversals for E-semiabundant semigroups and obtain some basic properties of E-semiabundant semigroups containing a multiplicative restriction transver- sal. Then we provide a construction method for E-semiabundant semigroups containing a multiplicative restriction transversal by using the Munn semigroup of an admissible quadruple and a restriction semigroup under some natural conditions. Our construction is similar to Hall's spined product construction of an orthodox semigroup. As a corollary, we obtain a new construction of a regular semigroup with a multiplicative inverse transversal and an abundant semigroup having a multiplicative ample transversal, which enriches the corresponding results obtained by Blyth-McFadden and El-Qallali, respectively.

THE LATTICE OF VARIETIES OF STRICT LEFT RESTRICTION SEMIGROUPS

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety $\mathbf{B}$ of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety $\mathbf{B}_{R}$ of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices ${\mathcal{L}}(\mathbf{B})$ and ${\mathcal{L}}(\mathbf{B}_{R})$ of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval $[\mathbf{D},\mathbf{D}\vee \mathbf{M}]$, every subvariety of $\mathbf{B}$ is induced from a member of $\mathbf{B}_{R}$ and vice versa. Here $\mathbf{D}$ is generated by the three-element left restriction semigroup $D$ and $\mathbf{M}$ is the variety of monoids. The analogues hold for pseudovarieties.

VARIETIES OF LEFT RESTRICTION SEMIGROUPS

Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. They may be defined by a simple set of identities and the author initiated a study of the lattice of varieties of such semigroups, in parallel with the study of the lattice of varieties of two-sided restriction semigroups. In this work we study the subvariety $\mathbf{B}$ generated by Brandt semigroups and the subvarieties generated by the five-element Brandt inverse semigroup $B_{2}$, its four-element restriction subsemigroup $B_{0}$ and its three-element left restriction subsemigroup $D$. These have already been studied in the ‘plain’ semigroup context, in the inverse semigroup context (in the first two instances) and in the two-sided restriction semigroup context (in all but the last instance). The author has previously shown that in the last of these contexts, the behavior is pathological: ‘almost all’ finite restriction semigroups are inherently nonfinitely based. Here we show that this is not the case for left restriction semigroups, by exhibiting identities for the above varieties and for their joins with monoids (the analog of groups in this context). We do so by structural means involving subdirect decompositions into certain primitive semigroups. We also show that each identity has a simple structural interpretation.

THE SEMIGROUPS B2 AND B0 ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids". These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻. For example, explicit bases of identities are found for the varieties generated by B0 and B2.