Numerical semigroups with many gaps of the same parity

Let g _e (S) (respectively, g _o (S)) be the number of even (respectively, odd) gaps of a numerical semigroup S. In this work we study and characterize the numerical semigroups S that verify 2|g _e (S)−g _o (S)|+1∈S. As a consequence we will see that every numerical semigroup can be represented by means of a numerical semigroup with maximal embedding dimension with all its minimal generators odd. The first author is supported by the project MTM2007-62346 and FEDER funds. The authors want to thank P.A. García-Sánchez and the referee for their comments and suggestions.     

Numerical Semigroups with a Monotonic Apery Set

We study numerical semigroups S with the property that if m is the multiplicity of S and w(i) is the least element of S congruent with i modulo m, then 0 < w(1) < ... < w(m − 1). The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups. This paper was supported by the project BFM2000-1469. The fourth author wishes to acknowledge support from the Universidade de Evora and the CIMA-UE.     

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)     

Weierstrass semigroups whose minimum positive integers are even

We consider three subsets of the set of 2n-semigroups, where for a positive integer n a 2n-semigroup means a numerical semigroup whose minimum positive integer is 2n. These three subsets are obtained by the Weierstrass semigroups of total ramification points on a cyclic covering of the projective line, the Weierstrass semigroups of ramification points on a double covering of a non-singular curve and the Weierstrass semigroups of points on a non-singular curve. We show that the three subsets are different for n ≧ 3. Partially supported by Grant-in-Aid for Scientific Research (17540046), Japan Society for the Promotion of Science. Received: 19 June 2006     

Completely simple semigroups with nilpotent structure groups

In this paper we find simple characterizations of completely simple semigroups with H-classes nilpotent of class ≤c, and of completely simple semigroups whose core has H-classes nilpotent of class ≤c. The notion of w-marginal completely regular semigroups is introduced, generalizing the concept of central semigroups. A law characterizing [x ₁,x ₂,…,x _c+1]-marginal completely simple semigroups is obtained. Additionally, the least congruences corresponding to these classes are described. Our results extend the corresponding results obtained by Petrich and Reilly in the abelian case. The author was supported by the Ministry of Higher Education, Science and Technology of Slovenia.     

Super rpp semigroups

We study a class of special strongly rpp semigroups, namely, the class of super rpp semigroups. These super rpp semigroups are generalizations of both superabundant semigroups and Clifford semigroups within the class of rpp semigroups. In particular, we prove that a super rpp semigroup is a semilattice of D ^(l)-simple strongly rpp semigroups. Our result not only generalizes a well-known theorem of Clifford in the class of completely regular semigroups but also strengthens some structure theorems obtained by Ren-Shum for superabundant semigroups which are orthodox. Some special super rpp semigroups are considered and discussed.     

Regular F-Abundant Semigroups

ABSTRACT

The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups.     

Irreducible numerical semigroups with multiplicity three and four

In this paper we analyze the irreducibility of numerical semigroups with multiplicity up to four. Our approach uses the notion of Kunz-coordinates vector of a numerical semigroup recently introduced in Blanco and Puerto (SIAM J. Discrete Math., 26(3):1210–1237, 2012). With this tool we also completely describe the whole family of minimal decompositions into irreducible numerical semigroups with the same multiplicity for this set of numerical semigroups. We give detailed examples to show the applicability of the methodology and conditions for the irreducibility of well-known families of numerical semigroups such as those that are generated by a generalized arithmetic progression.     

Basic properties of e-inversive semigroups

Generalizing a property of regular resp. finite semigroups a semigroup S is called E-(0-) inversive if for every a ∈ S4(a ≠ 0) there exists x ∈ S such that ax (≠ 0) is an idempotent. Several characterizations are given allowing to identify the (completely, resp. eventually) regular semigroups in this class. The case that for every a ∈ S4(≠ 0) there exist x,y ∈ S such that ax = ya(≠ 0) is an idempotent, is dealt with also. Ideal extensions of E- (0-)inversive semigroups are studied discribing in particular retract extensions of completely simple semigroups. The structure of E- (0-)inversive semigroups satisfying different cancellativity conditions is elucidated. 1991 AMS classification number: 20M10.     

Definability in the lattice of equational theories of semigroups

We study first-order definability in the latticeL of equational theories of semigroups. A large collection of individual theories and some interesting sets of theories are definable inL. As examples, ifT is either the equational theory of a finite semigroup or a finitely axiomatizable locally finite theory, then the set {T, T ^ϖ} is definable, whereT ^ϖ is the dual theory obtained by inverting the order of occurences of letters in the words. Moreover, the set of locally finite theories, the set of finitely axiomatizable theories, and the set of theories of finite semigroups are all definable. The research of both authors was supported by National Science Foundation Grant No. DMS-8302295     

Globals of Idempotent Semigroups

If S is a semigroup, the global (or the power semigroup) of S is the set P(S) of all nonempty subsets of S equipped with a naturally defined multiplication. A class K of semigroups is globally determined if any two semigroups of K with isomorphic globals are themselves isomorphic. We study properties of globals of idempotent semigroups and show, in particular, that the class of normal bands is globally determined.     

Affine convex body semigroups

In this paper we present a new class of semigroups called convex body semigroups which are generated by convex bodies of ? ^k . They generalize to arbitrary dimension the concept of proportionally modular numerical semigroups of Rosales et al. (J. Number Theory 103, 281–294, 2003). Several properties of these semigroups are proven. Affine convex body semigroups obtained from circles and polygons of ?² are characterized. The algorithms for computing minimal system of generators of these semigroups are given. We provide the implementation of some of them.     

Inverse semigroups and combinatorial C*-algebras

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case. *Partially supported by CNPq.     

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.     

Generation of analytic semigroups in theL p topology by elliptic operators inR n

Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate analytic semigroups of linear operators onL ^p(R ⁿ ), 1≦p≦∞. An explicit characterization of the domain is given for 1<p<∞. An application to parabolic problems is also included. This work has been partially supported by the Research Funds of the Ministero della Pubblica Istruzione. The authors are members of GNAFA (Consiglio Nazionale delle Ricerche).     

Proportionally modular diophantine inequalities and their multiplicity

Let I be an interval of positive rational numbers. Then the set S （I） = T ∩ N, where T is the submonoid of （Q0＋, ＋） generated by T, is a numerical semigroup. These numerical semigroups are called proportionally modular and can be characterized as the set of integer solutions of a Diophantine inequality of the form ax rood b 〈 cx. In this paper we are interested in the study of the maximal intervals I subject to the condition that S （I） has a given multiplicity. We also characterize the numerical semigroups associated with these maximal intervals.     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Minimal isometric dilations of operator cocycles

We obtain existence, uniqueness results for minimal isometric dilations of contractive cocycles of semigroups of unital *-endomorphisms ofB(H. This generalizes the result of Sz. Nagy on minimal isometric dilations of semigroups of contractive operators on a Hilbert space. In a similar fashion we explore results analogus to Sarason's characterization that subspaces to which compressions of semigroups are again semigroups are semi-invariant subspaces, in the context of cocycles and quantum dynamical semigroups.This research is supported by the Indian National Science Academy under Young Scientist Project.