Dimonoids

It is proved that a system of axioms for a dimonoid is independent and Cayley’s theorem for semigroups has an analog in the class of dimonoids. The least separative congruence is constructed on an arbitrary dimonoid endowed with a commutative operation. It is shown that an appropriate quotient dimonoid is a commutative separative semigroup. The least separative congruence on a free commutative dimonoid is characterized. It is stated that each dimonoid with a commutative operation is a semilattice of Archimedean subdimonoids, each dimonoid with a commutative periodic semigroup is a semilattice of unipotent subdimonoids, and each dimonoid with a commutative operation is a semilattice of a-connected subdimonoids. Various dimonoid constructions are presented.     

Automorphisms of the endomorphism semigroup of a free commutative dimonoid

We determine all isomorphisms between the endomorphism semigroups of free commutative dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative dimonoid are quasi-inner. In particular, we answer a question of B. I. Plotkin.     

Structure of relatively free dimonoids

Loday introduced the notion of a dimonoid and constructed the free dimonoid. The paper concerns the variety theory of dimonoids. We recall and summarize the results obtained by Loday, the author as well as others on the structure of some relatively free dimonoids. This part of the paper should be viewed as a survey, however we include some new ideas in the last sections. Namely, we construct a free (?z;rs)-dimonoid, a free (rs;rz)-dimonoid, a free rs-dimonoid, a free (rb;rs)-dimonoid, a free (rs;rb)-dimonoid and characterize the least (?z;rs)-congruence, the least (rs;rz)-congruence, the least rs-congruence, the least (rb;rs)-congruence and the least (rs;rb)-congruence on the free dimonoid. We also establish that the automorphism groups of constructed free algebras are isomorphic to the symmetric group and the semigroups of the free rs-dimonoid are anti-isomorphic.     

Certain congruences on free trioids

Loday and Ronco introduced the notion of a trioid and constructed the free trioid of rank 1. This paper is devoted to the study of congruences on trioids. We characterize the least dimonoid congruences and the least semigroup congruence on the free (commutative, rectangular) trioid.     

Free dimonoids

We characterize the least semilattice congruence of a free dimonoid and prove that a free dimonoid is a semilattice of s-simple subdimonoids each of which is a rectangular band of subdimonoids.     

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.     

Finite state wreath powers of transformation semigroups

The finite state wreath power of a transformation semigroup is introduced. It is proved that the finite state wreath power of nontrivial semigroup is not finitely generated and in some cases even does not contain irreducible generating systems. The free product of two monogenic semigroups of index 1 and period m is constructed in the finite state wreath power of corresponding monogenic monoid.     

On compact quantum semigroup QS red

We study some properties of a reduced semigroup C*-algebra of a semigroup S. For the semigroup C*-algebra generated by the deformation of the algebra of continuous functions on a compact abelian group we obtain a structure of a compact quantum semigroup. We also consider morphisms of constructed compact quantum semigroups.     

具有逆断面的局部逆半群

Let S be a locally inverse semigroup with an inverse transversal S°. In this paper, we construct an amenable partial order on S by an R-cone. Conversely, every amenable partial order on S can be constructed in this way. We give some properties of a locally inverse semigroup with a Clifford transversal. In particular, if S is a locally inverse semigroup with a Clifford transversal, then there is an order-preserving bijection from the set of all amenable partial orders on S to the set of all R-cones of S.     

Guarded and Banded Semigroups

The variety of guarded semigroups consists of all (S,·, ˉ) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup S contains a canonical subsemilattice g_⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g_⋆ ≅ L is obtained.     

Structure of free strong doppelsemigroups

We consider strong doppelsemigroups which are sets with two binary associative operations satisfying axioms of strong interassociativity. Commutative dimonoids in the sense of Loday are examples of strong doppelsemigroups and two strongly interassociative semigroups give rise to a strong doppelsemigroup. The main aim of this paper is to construct a free strong doppelsemigroup, a free n-dinilpotent strong doppelsemigroup, a free commutative strong doppelsemigroup and a free n-nilpotent strong doppelsemigroup. We also characterize the least n-dinilpotent congruence, the least commutative congruence, the least n-nilpotent congruence on a free strong doppelsemigroup and establish that the automorphism group of every constructed free algebra is isomorphic to the symmetric group.     

Proper Covers for Left Ample Semigroups

A left ample semigroup is a semigroup with a unary operation ⁺ which has a (2,1)-algebra embedding into a symmetric inverse monoid I(X), the operation ⁺ on I(X) being defined by α⁺ = αα^-1. We consider some analogues for left ample semigroups of results on E-unitary covers of inverse semigroups due to McAlister and Reilly. The analogue of an E-unitary cover is a proper cover, and we discuss the construction of proper covers in terms of relational homomorphisms, and of dual prehomomorphisms. We observe that our construction gives an E-dense proper cover for an E-dense left ample semigroup. We also consider proper covers constructed from strict embeddings into factorisable left ample monoids. In contrast to the inverse case, not all proper covers arise in this way. However, in the E-dense case, we characterise those E-dense proper covers which can be constructed from such embeddings.     

Ternary semigroups

A ternary semigroup is a nonempty set together with a ternary multiplication which is associative. Analogous to the theory of semigroups, a regularity condition on a ternary semigroup is introduced and the properties of regular ternary semigroups are studied. Associated with a ternary semigroup, a semigroup called the semigroup cover is constructed and its properties are investigated.     

平衡范畴与半群的双序

研究范畴与半群通过幂等元双序建立的一种自然联系.对每个有幂等元的半群S,其幂等元生成的左、右主理想之集通过双序ω~e,ω~r自然确定两个有子对象、有像且每个包含都右可裂的范畴L(S),R(S),其中态射的性质与S中元素的富足性、正则性有自然对应.利用这个联系,我们定义了"平衡(富足、正规)范畴"概念.对任一平衡(富足、正规)范畴■,我们构造其"锥半群"■,证明■左富足(富足、正则),且每个平衡(富足、正规)范畴■都与某左富足(富足、正则)半群S的左主理想范畴L(S)(作为有子对象的范畴)同构.     

Generators for categories of compact irreducible semigroups

For a compact totally ordered space X, K. H. Hofmann and P. S. Mostert constructed a topological semigroup Irr (X)₀ such that every compact irreducible semigroup with idempotents X is a surmorphic image of Irr (X)₀ [2]. J. H. Carruth and M. Mislove pointed out that Irr (X)₀ was in general not a compact semigroup. In this paper a compact connected hormos will be constructed which contains Irr (X)₀ as a dense subsemigroup and for which every compact irreducible semigroup with idempotents X is a surmorphic image. This leads to a new proof of the existence of generators for the category of compact irreducible semigroups with idempotents X. It will then be shown that Irr (X)₀ contains generators for the category.     

Matrix rank 1 semigroup identities

A finite basis of identities is constructed for the semigroup of all rank 1 n × n matri­ces over the field. It is worthy to notice that every semigroup of all rank r, r > l,n×n matrices over a finite field has no finite basis of identities. Let G be an arbitrary vari­ety of groups with a finite basis of identities. A finite basis of identities is constructed for the variety generated by all completely 0-simple semigroups over G-groups.     

P-正则半群的结构

本文定义了一类称之为PR-系的代数,证明了每个PR-系是一个P-正则半群, 而且每个P-正则半群可用这种方式来进行构造.     

Non-Differentiable Skew Convolution Semigroups and Related Ornstein–Uhlenbeck Processes

It is proved that a general non-differentiable skew convolution semigroup associated with a strongly continuous semigroup of linear operators on a real separable Hilbert space can be extended to a differentiable one on the entrance space of the linear semigroup. A càdlàg strong Markov process on an enlargement of the entrance space is constructed from which we obtain a realization of the corresponding Ornstein–Uhlenbeck process. Some explicit characterizations of the entrance spaces for special linear semigroups are given.     

The structure of L*-inverse semigroups

The concepts of L*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the L*-inverse semigroup can be described as the left wreath product of a type A semigroupΓand a left regular band B together with a mapping which maps the semigroupΓinto the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the L*-inverse semigroups by using the left wreath products.     

Quotient space of LMC-compactification as a space of z-filters

The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup. In this paper an internal construction of a quotient space of the left multiplicative continuous compactification of a semitopological semigroup is constructed as a space of z-filters.