Malcev presentations for subsemigroups of direct products of coherent groups

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free subgroup of finite index.) By considering the direct product of two free semigroups, it is also shown that polycyclic groups, unlike nilpotent groups, can contain finitely generated subsemigroups that do not admit finite Malcev presentations.     

*-orderable semigroups

Fix a *-orderable field k. We introduce the class of *-orderable semigroups as those semigroups with involution S for which the semigroup algebra kS endowed with the canonical involution admits a *-ordering. It is shown that this class is a quasivariety that is locally and residually closed. A cancellative nilpotent semigroup with involution is proved to be *-orderable if and only if it has unique extraction of roots. In general this equivalence fails, although every *-orderable semigroup has unique extraction of roots.     

On semigroups admitting ring structure

A multiplicative semigroup S with 0 is said to be a R-semigroup if S admits a ring structure. Isbell proved that if a finitely generated commutative semigroup is a R-semigroup, then it should be finite. The non-commutative version of this theorem is unsettled. This paper considers semigroups, not necessarily commutative, which are principally generated as a right ideal by single elements and semigroups which are generated by two independent generators and describes their structure. We also prove that if a cancellative 0-simple semigroup containing an identity is a R-semigroup, then it should be a group with zero. Communicated by A. H. Clifford     

Paradoxical Decomposition of Semigroups

We introduce the concept of paradoxical decomposition for semigroups. We show that a semigroup S admits a left paradoxical decomposition if and only if S is not left amenable. Also Rosenbelatt and Willis introduced the configuration concept for groups and showed that amenability of groups is equivalent with existence of normalised solution for any system of configuration equations. We generalise it for semigroups.     

SEMIGROUPS CHARACTERIZED BY THEIR FUZZY IDEALS

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Square roots and continuity in strictly linearly ordered semigroups on real intervals

In this article we show that the semigroup operation of a strictly linearly ordered semigroup on a real interval is automatically continuous if each element of the semigroup admits a square root. Hence, by a result of Aczél, such a semigroup is isomorphic to an additive subsemigroup of the real numbers.     

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.     

Free Completely J~(e)-simple Semigroups

A semigroup is called completely J~((e))-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid.It is proved that completely J~((e))-simple semigroups form a quasivarr ity.Moreover,the construction of free completely J~((e))-simple semigroups is given.It is found that a free completely J~((e))-simple semigroup is just a free completely J~*-simple semigroup and also a full subsemigroup of some completely simple semigroups.     

On finite-dimensional representations of compact inverse semigroups

W.D. Munn proved that a finite-dimensional representation of an inverse semigroup is equivalent to a ?-representation (by partial isometries) if and only if it is bounded. This paper gives a new analytic proof that every finite-dimensional representation of a compact inverse semigroup is equivalent to a ?-representation. This will be the main result of this paper.     

Free completely <Emphasis Type="Italic">J</Emphasis><Superscript>(<Emphasis Type="Italic">ℓ</Emphasis>)</Superscript>-simple Semigroups

A semigroup is called completely J^(ι)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J^(ι)-simple semigroups form a quasivarity. Moreover, the construction of free completely J^(ι)-simple semigroups is given. It is found that a free completely J^(ι)-simple semigroup is just a free completely J ^*-simple semigroup and also a full subsemigroup of some completely simple semigroups.     

Positive reynolds operators and generating derivations

It is shown that the spectrum of a positive Reynolds operator on C₀(X) is contained in the disc centered at 1/2 with radius 1/2. Moreover, every positive Reynolds operator T with dense range is injective. In this case, the operator D = 1 — T^?1 is a densely defined derivation, which generates a one — parameter semigroup of algebra homomorphisms. This semigroup yields an integral representation of T. Along the way, it is proved that a densely defined closed derivation D generates a semigroup if, and only if, R(1, D) exists and is a positive operator.     

所有双理想都是素双理想的半群

设S为一个半群,B是S的一个双理想(即B是S的一个满足条件BSBB的子半群)．如果对S的任意双理想C,D都有CDB蕴涵CB或DB,我们就称B为S的一个素双理想．如果K是一个N-覆盖的纯LR-带,我们就称K为一条拟链．本文证明了半群S的所有双理想都是素双理想的充分必要条件是S是一个幂等元形成拟链的纯整群并半群．     

On finitely generated subsemigroups of a free semigroup

A finitely generated semigroup S given by generators and the set of relations between these generators is called an F-semigroup if it is isomorphic to a subsemigroup of a free semigroup. In this paper three theorems concerning F-semigroups are proven and some unsolved problems are presented.     

On free spectra of finite completely regular semigroups and monoids

We show that a finite completely regular semigroup has a sub-log-exponential free spectrum if and only if it is locally orthodox and has nilpotent subgroups. As a corollary, it follows that the Seif Conjecture holds true for completely regular monoids. In the process, we derive solutions of word problems of free objects in a sequence of varieties of locally orthodox completely regular semigroups from solutions of word problems in relatively free bands.     

Semisimplicity and von Neumann's regularity

The purpose of this paper is to link semigroup theory and category theory by using Green's relations to study regular morphisms and semisimple objects in a category which is filtering in the sense that every morphism admits an epic-monic factorization. For example, the following results are shown for a filtering category C ∶ (1)C is regular if and only if every monic inC is left invertible and every epic inC is right invertible: (2) the monoid of endomorphisms of any semisimple object ofC is regular. Communicated by A.H. Clifford     

On F-inverse covers of inverse monoids

It is shown that each finite inverse monoid admits a finite F-inverse cover if and only if the same is true for each finite combinatorial strict inverse semigroup with an identity adjoined if and only if the same is true for the Margolis-Meakin expansion M(H) of each finite elementary abelian p-group H for some prime p. Additional equivalent conditions are given in terms of the existence of locally finite varieties of groups having certain properties. Ultimately, the problem of whether each finite inverse monoid admits a finite F-inverse cover, is reduced to a question concerning the Kostrikin-Zelmanov varieties K_n of all locally finite groups of exponent dividing n.     

Every finite semigroup is embeddable in a finite relatively free semigroup

The title result is proved by a Murskii-type embedding.Results on some related questions are also obtained. For instance, it is shown that every finitely generated semigroup satisfying an identity ξ^d=ξ^2d is embeddable in a relatively free semigroup satisfying such an identity, generally with a larger d; but that an uncountable semigroup may satisfy such an identity without being embeddable in any relatively free semigroup.It follows from known results that every finite group is embeddable in a finite relatively free group. It is deduced from this and the proof of the title result that a finite monoid S is embeddable by a monoid homomorphism in a finite (or arbitrary) relatively free monoid if and only if its group of invertible elements is either {e} or all of S.     

The Similarity Problem for Bounded Analytic Semigroups on Hilbert Space

t )_t≥0 on a Hilbert space H, we establish conditions under which (T_t)_t≥0 is similar to a contraction semigroup, i.e., there exists an isomorphism S Ε B (H) such that (S^-1 T_t S)_t≥0 is a contraction semigroup. In the case when the generator -A of (T_t)_t≥0 is one-to-one, we obtain that (T_t)_t≥0 is similar to a contraction semigroup if and only if A admits bounded imaginary powers. This characterizes one-to-one operators of type strictly less than π/2 on H which belong to BIP (H).     

Split Left GC-Lpp Semigroups

A left GC-lpp semigroup S is called split if the natural homomorphism γb of S onto S/γ induced by γ is split.It is proved that a left GC-lpp semigroup is split if and only if it has a left adequate transversal.In particular,a construction theorem for split left GC-lpp semigroups is established.