A Characterisation of a Class of Semigroups with Locally Commuting Idempotents

McAlister proved that a necessary and sufficient condition for a regular semigroup S to be locally inverse is that it can be embedded as a quasi-ideal in a semigroup T which satisfies the following two conditions: (1) T = TeT, for some idempotent e; and (2) eTe is inverse. We generalise this result to the class of semigroups with local units in which all local submonoids have commuting idempotents.     

On <Emphasis Type="Italic">U</Emphasis>-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W _U and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups. This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123)     

Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.     

Chapter 1. Extension of the fundamental theorem of finite semigroups

This paper proves that some useful commutivity relations exist among semigroup wreath product factors that are either groups or combinatorial “units” U₁, U₂, or U₃. Using these results it then obtains some characterizations of each of the classes of semigroups buildable from U₁'s, U₂'s, and groups (“buildable” meaning “dividing a wreath product of”).We show that up to division U₁'s can be moved to the right and U₂'s, and groups to the left over other units and groups, if it is allowed that the factors involved be replaced by their direct products, or in the case of U₂, even by a wreath product. From this it is deduced that U₁'s and U₂'s do not affect group complexity, that any semigroup buildable from U₁'s, U₂'s, and groups has group complexity 0 or 1, and that all such semigroups can be represented, up to division, in a canonical form—namely, as a wreath product with all U₁'s on the right, all U₂'s on the left, and a group in the middle. This last fact is handy for developing charactérizations.An embedding theorem for semigroups with a unique 0-minimal ideal is introduced, and from this and the commutivity results and some constructions proved for RLM semigroups, there is obtained an algebraic characterization for each class of semigroups that is a wreath product-division closure of some combination of U₁'s, U₂'s, and the groups. In addition it is shown, for i = 1,2,3, that if the unit U_i does not divide a semigroup S, then S can be built using only groups and units not containing U_i. Thus, it can be deduced that any semigroup which does not contain U₃ must have group complexity either 0 or 1. This then establishes that indeed U₃ is the determinant of group complexity, since it is already proved that both U₁ and U₂ are transparent with regard to the group complexity function, and it is known that with U₃ (and groups) one can build semigroups with complexities arbitrarily large. Another conclusion is a combinatorial counterpart for the Krohn-Rhodes prime decomposition theorem, saying that any semigroups can be built from the set of units which divide it together with the set of those semigroups not having unit divisors. Further, one can now characterize those semigroups which commute over groups, showing a semigroup commutes to the left over groups iff it is “R₁” (i.e., does not contain U₁, i.e., is buildable form U₂'s and groups), and commutes to the right over groups iff it does not contain U₂ (i.e., is buildable from groups and U₁'s). Finally, from the characterizations and their proofs one sees some ways in which groups can do the work of combinatorials in building combinatorial semigroups.     

Sulľapplicazione della funzione ellitticap u alla teoria dei poligoni di poncelet

Abstract  In this paper we study strongly continuous positive semigroups on particular classes of weighted continuous function space on a locally compact Hausdorff space X having a countable base. In particular we characterize those positive semigroups which are the transition semigroups of suitable Markov processes. Some applications are also discussed. Keywords Positive semigroup, Markov transition function, Markov process, Weighted continuous function space, Degenerate second order differential operator Mathematics Subject Classification (2000) 47D06, 47D07, 60J60     

A finiteness lemma,brauer's theorem and other iIrreducibility results

A technical lemma is proved for certain semigroups of matrices. It has several applications to problems concerning irreducible semigroups satisfying spectral conditions, e.g., submultiplicativity of spectrum. It is also used to give extensions of the following theorem of Brauer's. If U is a finite group of complex matrices, so that for some integer m, every U in U, satisties U^m =I then U has a representation over the cyclotomic field Q(ω), where ω is a primitive m-th root of unity.     

The structure of medial weakly U-abundant semigroups

We consider the class of weakly U-abundant semigroups satisfying the congruence condition (C) containing both the class of regular semigroups and the class of abundant semigroups as its subclasses. The class of weakly U-abundant semigroups with a medial projection satisfying the congruence condition (C) will be particularly studied. This kind of semigroups will be called medial weakly U-abundant semigroups. In this paper, we establish a structure theorem for such semigroups. It is proved that every medial weakly U-abundant semigroup can be expressed by some kind of bands and quasi-Ehresmann semigroups. Our theorem generalizes and enriches the structure theorem given by M. Loganathan in 1987 for regular semigroups with a medial idempotent.     

Variants of Regular Semigroups

Let S be a regular semigroup, and let a ∈ S . Then a variant of S with respect to a is a semigroup with underlying set S and multiplication \circ defined by x \circ y = xay . In this paper, we characterise the regularity preserving elements of regular semigroups; these are the elements a such that (S,\circ) is also regular. Hickey showed that the set of regularity preserving elements can function as a replacement for the unit group when S does not have an identity. As an application, we characterise the regularity preserving elements in certain Rees matrix semigroups. We also establish connections with work of Loganathan and Chandrasekaran, and with McAlister's work on inverse transversals in locally inverse semigroups. We also investigate the structure of arbitrary variants of regular semigroups concentrating on how the local structure of a semigroup affects the structure of its variants. May 24, 1999     

Operators on the Lattice of Pseudovarieties of Finite Semigroups

L (F) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, Auinger, Hall and the present authors recently introduced fourteen complete congruences on L (F). Such congruences provide a framework from which to study L (F) both locally and globally. For each such congruence ρ and each U∈L (F) the ρ-class of U is an interval [U ^ρ, U _ρ]. This provides a family of operators of the form U⇒Uρ on L (F) that reveal important relationships between elements of L (F). Various aspects of these operators are considered including characterizations of U ^ρ, bases of pseudoidentities for U ^ρ, instances of commutativity (U ^ρ)^σ = U ^σ)^ρ, as well as the semigroups generated by certain pairs of such operators.     

The Universal Covering of an Inverse Semigroup

We examine an inverse semigroup T in terms of the universal locally constant covering of its classifying topos . In particular, we prove that the fundamental group of coincides with the maximum group image of T. We explain the connection between E-unitary inverse semigroups and locally decidable toposes, characterize E-unitary inverse semigroups in terms of a kind of geometric morphism called a spread, characterize F-inverse semigroups, and interpret McAlister’s “P-theorem” in terms of the universal covering.     

On the decidability of the word problem for amalgamated free products of inverse semigroups

We study inverse semigroup amalgams [S ₁,S ₂;U], where S ₁ and S ₂ are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S ₁ and S ₂. We use a modified version of Bennett’s work on the structure of Schützenberger graphs of the ℛ-classes of S ₁* _U S ₂ to state sufficient conditions for the amalgamated free products S ₁* _U S ₂ having decidable word problem.     

On finiteness conditions for Rees matrix semigroups

Let T=[S; I; J; P] be a Rees matrix semigroup where S is a semigroup, I and J are index sets, and P is a J × I matrix with entries from S, and let U be the ideal generated by all the entries of P. If U has finite index in S, then we prove that T is periodic (locally finite) if and only if S is periodic (locally finite). Moreover, residual finiteness and having solvable word problem are investigated.     

Perfect Semigroups and Aliens

Abstract. A topologized semigroup is called perfect if its multiplication is a perfect map (= a closed continuous mapping such that the inverse image of every point is compact). Thus a locally compact topological semigroup is perfect if and only if its multiplication is closed and each of its elements is compactly divided , that is, its divisors form a compact set. In the present paper we study compactly and non-compactly divided elements in the contexts of general locally compact semigroups, subsemigroups of groups, Lie semigroups and subsemigroups of Sl(2,R).     

Convolution semigroups of states

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C ₀-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.     

Idempotent probabilities on semigroups

Summary In this paper, idempotent probability measures have been considered on semigroups which are locally compact or metric and satisfy: (*) A ^–1 B and Ax ^–1 are compact whenever A and B are so, for every x in the semigroup. Such semigroups are more general than compact semigroups which do admit of such measures. On such semigroups we can construct such measures by the usual process if there is a compact sub-semigroup. It is shown in this paper that if such a measure exists in such semigroups, then it must be such an extension measure. Some related results concerning the conditions (*) are also discussed here.     

On Incidence Algebras of Combinatorial Inverse Monoids

We study the incidence algebra of the reduced standard division category of a combinatorial bisimple inverse monoid [with (E(S), ≤) locally finite], and we describe semigroups of poset type (i.e., a combinatorial inverse semigroup for which the corresponding Möbius category is a poset) as being combinatorial strict inverse semigroups. Up to isomorphism, the only Möbius-division categories are the reduced standard division categories of combinatorial inverse monoids.