Continuous extension of a densely parameterized semigroup

Let be a dense sub-semigroup of ℝ₊, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over can be extended to a weakly continuous semigroup over ℝ₊. We obtain similar results for nonlinear, nonexpansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over ℝ₊. O.M. Shalit was partially supported by the Gutwirth Fellowship.     

A Representation for $F$-Regular Semigroups

A regular (inverse) semigroup S is called F-regular (F-inverse), if each class of the least group congruence _S contains a greatest element with respect to the natural partial order on S. Such a semigroup is necessarily an E-unitary regular (hence orthodox) monoid. We show that each F-regular semigroup S is isomorphic to a well determined subsemigroup of a semidirect product of a band X by S/_S, where X belongs to the band variety, generated by the band of idempotents E_S of S. Our main result, Theorem 4, is the regular version of the corresponding fact for inverse semigroups, and might be useful to generalize further features of the theory of F-inverse semigroups to the F-regular case.     

On algebras of <Emphasis Type="Italic">P</Emphasis>-Ehresmann semigroups and their associate partial semigroups

P-Ehresmann semigroups are introduced by Jones as a common generalization of Ehresmann semigroups and regular \(*\)-semigroups. Ehresmann semigroups and their semigroup algebras are investigated by many authors in literature. In particular, Stein shows that under some finiteness condition, the semigroup algebra of an Ehresmann semigroup with a left (or right) restriction condition is isomorphic to the category algebra of the corresponding Ehresmann category. In this paper, we generalize this result to P-Ehresmann semigroups. More precisely, we show that for a left (or right) P-restriction locally Ehresmann P-Ehresmann semigroup \(\mathbf{S}\), if its projection set is principally finite, then we can give an algebra isomorphism between the semigroup algebra of \(\mathbf{S}\) and the partial semigroup algebra of the associate partial semigroup of \(\mathbf{S}\). Some interpretations and necessary examples are also provided to show why the above isomorphism dose not work for more general P-Ehresmann semigroups.     

Semigroup Varieties on Whose Free Objects Almost All Fully Invariant Congruences are Weakly Permutable

A semigroup variety is said to be of index 2 if all nil-semigroups of the variety are semigroups with zero multiplication. We describe all semigroup varieties V of index 2 on free objects of which every two fully invariant congruences contained in the least semilattice congruence are weakly permutable, and semigroup varieties of index 2 all of whose subvarieties share the above-mentioned property.     

Markov Properties of Burnside Varieties of Semigroups

It is proved that every Markov property of semigroups finitely presented in a variety given by the identity x^r1=x^r2, where r₁>₂2, which a one-element semigroup enjoys, is algorithmically non-recognizable.     

Duals and topological center of a class of matrix algebras with applications

We characterize the topological center of a class of matrix algebras, which are called -Munn algebras. This involves a characterization of Arens regular semigroup algebras for a large class of semigroups, which is an extension of Young's Theorem for semigroups. We show by some counter examples that only up to a certain extent Young's Theorem can be generalized.

     

Tight representations of semilattices and inverse semigroups

By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleanness present in the semilattice of idempotents of  . After observing that the Vagner–Preston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the regular tight representation. We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to continuous inverse semigroups, a class generalizing the E ^*-unitaries. Partially supported by CNPq.     

Semigroups whose Proper Right Congruences Form a Tree

Let S be a semigroup whose set of proper right congruences form a tree. The main theorem is a characterization of those semigroups having this property. In this characterization we draw on the results of Schein and Tamura for commutative semigroups and Kozhukhov for left chain semigroups and Hitzel for nilpotent semigroups. The interested reader should also see the work of Nagy on -semigroups.     

Generalised domain and <Emphasis Type="Italic">E</Emphasis>-inverse semigroups

A generalised D-semigroup is here defined to be a left E-semiabundant semigroup S in which the \(\overline{\mathcal R}_E\)-class of every \(x\in S\) contains a unique element D(x) of E, made into a unary semigroup. Two-sided versions are defined in the obvious way in terms of \(\overline{\mathcal R}_E\) and \(\overline{\mathcal L}_E\). The resulting class of unary (bi-unary) semigroups is shown to be a finitely based variety, properly containing the variety of D-semigroups (defined in an order-theoretic way in Communications in Algebra, 3979–4007, 2014). Important subclasses associated with the regularity and abundance properties are considered. The full transformation semigroup \(T_X\) can be made into a generalised D-semigroup in many natural ways, and an embedding theorem is given. A generalisation of inverse semigroups in which inverses are defined relative to a set of idempotents arises as a special case, and a finite equational axiomatisation of the resulting unary semigroups is given.     

Conditions for Admissibility of Observation Operators and Boundedness of Hankel Operators

We derive new necessary and sufficient conditions for admissibility of observation operators for certain C ₀-semigroups. We also prove a new sufficient criterion for admissibility for observation operators with infinite-dimensional output space on contraction semigroups. If the contraction semigroup is completely non-unitary and its co-generator has finite defect indices, then this criterion is also necessary. In the case of the right shift semigroup on L ²(0,), these conditions translate into conditions for the boundedness of Hankel operators.     

Free <Emphasis Type="Italic">n</Emphasis>-Tuple Semigroups

The present paper is devoted to the study of n-tuple semigroups. A free n-tuple semigroup of arbitrary rank is constructed and, as a consequence, singly generated free n-tuple semigroups are characterized. Moreover, examples of n-tuple semigroups are presented, the independence of the n-tuple semigroup axioms is proved, and it is shown that the natural semigroups of the constructed free n-tuple semigroup are isomorphic and the automorphism group of this n-tuple semigroup is isomorphic to a symmetric group.     

Subdirectly irreducible \mathcal{RGC} -commutative right H-semigroups

A semigroup S is said to be ℛ-commutative if, for all elements a,b∈S, there is an element x∈S ¹ such that ab=bax. A semigroup S is called a generalized conditionally commutative (briefly, -commutative) semigroup if it satisfies the identity aba ²=a ² ba. An ℛ-commutative and -commutative semigroup is called an -commutative semigroup. A semigroup S is said to be a right H-semigroup if every right congruence of S is a congruence of S. In this paper we characterize the subdirectly irreducible semigroups in the class of -commutative right H-semigroups. Research supported by the Hungarian NFSR grant No T029525.     

On nearly abelian polynomial semigroups

Let be a polynomial semigroup containing an element with degree at least 2 with the semigroup operation being functional composition. We prove that is nearly abelian if and only if the smallest completely invariant closed subset of the Riemann sphere is not equal to the Riemann sphere. We also give a positive answer to Conjecture 7.1 in Hinkkanen and Martin's paper on the dynamics of semigroups of rational functions.

     

Euler’s Exponential Formula for Semigroups

The aim of this paper is to show that Eulers exponential formula $\lim_{n\rightarrow\infty}\linebreak[4] (I-tA/n)^{-n}x = e^{tA}x$, well known for $C_0$ semigroups in a Banach space $X\ni x$, can be used for semigroups not of class $C_0$, the sense of the convergence being related to the regularity of the semigroup for $t>0$. Although the strong convergence does not hold in general for not strongly continuous semigroups, an integrated version is stated for once integrated semigroups. Furthermore by replacing the initial topology on $X$ by some (coarser) locally convex topology $\tau$, the strong $\tau$-convergence takes place provided the semigroup is strongly $\tau$-continuous; in particular this applies to the class of bi-continuous semigroups. More generally if a $k$-times integrated semigroup $S(t)$ in a Banach space $X$ is strongly $k$-times $\tau$-differentiable, then Eulers formula holds in this topology with limit $S^{(k)}(t)$. On the other hand, for bounded holomorphic semigroups not necessarily of class $C_0$, Eulers formula is shown to hold in operator norm, with the error bound estimate ${\cal O}(\ln n/n)$, uniformly in $t>0$. All these results also concern degenerate semigroups.     

Subdirect products of regular semigroups

A nonempty subset X contained in anH-class of a regular semigroup S is called agroup coset in S if XX′X=X and X′XX′=X′ where X′ is the set of inverses of elements of X contained in anH-class of S. Let μ denote the maximum idempotent separating congruence on S. We show in Section 1 of this paper that the set K(S) of group cosets in S contained in the μ-classes of S is a regular semigroup with a suitably defined product. In Section 2, we describe subdirect products of twoinductive groupoids in terms of certain maps called ‘subhomomorphisms’. A special class of subdirect products, called S^*-direct products, is described in Section 3. In the remaining two sections, we give some applications of the construction of S^*-direct products for describing coextensions of regular semigroups and for providing a covering theorem for pseudo-inverse semigroups.     

\mathcal{P}\mathcal{A}{\text{-isomorphisms}} of inverse semigroups

A partial automorphism of a semigroup S is any isomorphism between its subsemigroups, and the set all partial automorphisms of S with respect to composition is an inverse monoid called the partial automorphism monoid of S. Two semigroups are said to be if their partial automorphism monoids are isomorphic. A class of semigroups is called if it contains every semigroup to some semigroup from Although the class of all inverse semigroups is not we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is A semigroup is called if it is isomorphic or antiisomorphic to any semigroup that is to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are To Ralph McKenzieReceived April 15, 2004; accepted in final form October 7, 2004.     

Left Orders in Special Completely 0-Simple Semigroups

A subsemigroup S of a semigroup Q is a left order in Q, and Q is a semigroup of left quotients of S, if every element of Q can be written as a ^?1 b for some ${a, b\in S}$ with a belonging to a group ${\mathcal{H}}$ -class of Q. Characterizations are provided for semigroups which are left orders in completely 0-simple semigroups in the following classes: without similar ${\mathcal{L}}$ -classes, without contractions, ${\mathcal{R}}$ -unipotent, Brandt semigroups and their generalization. Complete discussion of two examples and an idea for a new concept conclude the paper.     

Maximal semigroups in semi-simple Lie groups

The maximal semigroups with nonempty interior in a semi-simple Lie group with finite center are characterized as compression semigroups of subsets in the flag manifolds of the group. For this purpose a convexity theory, called here -convexity, based on the open Bruhat cells is developed. It turns out that a semigroup with nonempty interior is maximal if and only if it is the compression semigroup of the interior of a -convex set.

     

On Ordered Semigroups Which are Semilattices of Simple and Regular Semigroups

We characterize the ordered semigroups which are decomposable into simple and regular components. We prove that each ordered semigroup which is both regular and intra-regular is decomposable into simple and regular semigroups, and the converse statement also holds. We also prove that an ordered semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subsemigroup of S. An ordered semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subsemigroups of S. We characterize the chains of simple and regular semigroups, and we prove that S is a complete semilattice of simple and regular semigroups if and only if S is a semilattice of simple and regular semigroups. While a semigroup which is both π-regular and intra-regular is a semilattice of simple and regular semigroups, this does not hold in ordered semigroups, in general.     

Ideals and Representations of Certain Semigroup Algebras

We characterize various types of ideals in ¹-Munn algebras with a bounded approximate identity, in terms of the ideals of their underlying algebras. Moreover, we show the correspondence between different types of *-representations of ¹-Munn algebras and *-representations of their underlying algebras. In the context of semigroup algebras, .rst we prove the conjecture which was proposed in [8]. Then we apply the results of the first part to build various representations of unital semigroup algebras of completely 0-simple inverse semigroups. This generalizes some of the Barnes results [1] with different and much shorter proofs.