Matrices of bisimple regular semigroups

A semigroup S is a matrix of subsemigroups S_iμ, i ε I, μ ε M if the S_iμ form a partition of S and S_iμS_jν≤S_iν for all i, j in I, μ, ν in M. If all the S_iμ are bisimple regular semigroups, then S is a bisimple regular semigroup. Properties of S are considered when the S_iμ are bisimple and regular; for example, if S is orthodox then each element of S has an inverse in every component S_iμ.     

Semilattices of bisimple orthodox semigroups

A structure theorem for bisimple orthodox semigroups was given by Clifford [2]. In this paper we determine all homomorphisms of a certain type from one bisimple orthodox semigroup into another, and apply the results to give a structure theorem for any semilattice of bisimple orthodox semigroups with identity in which the set of identity elements forms a subsemigroup. A special case of these results is indicated for bisimple left unipotent semigroups.     

Medial permutable semigroups of the first kind

We say that a semigroup S is a permutable semigroup if the congruences of S commute with each other, that is, α○β=β○α is satisfied for all congruences α and β of S. A semigroup is called a medial semigroup if it satisfies the identity axyb=ayxb. The medial permutable semigroups were examined in Proc. Coll. Math. Soc. János Bolyai, vol. 39, pp. 21–39 (1981), where the medial semigroups of the first, the second and the third kind were characterized, respectively. In Atta Accad. Sci. Torino, I-Cl. Sci. Fis. Mat. Nat. 117, 355–368 (1983) a construction was given for medial permutable semigroups of the second [the third] kind. In the present paper we give a construction for medial permutable semigroups of the first kind. We prove that they can be obtained from non-archimedean commutative permutable semigroups (which were characterized in Semigroup Forum 10, 55–66, 1975). Research supported by the Hungarian NFSR grant No T042481 and No T043034.     

On the translational hull of a type B semigroup

In this paper, the translational hull of a type B semigroup is considered. We prove that the translational hull of a type B semigroup is itself a type B semigroup, and give some properties and characterizations of the translational hulls of such semigroups. Moreover, we consider the translational hulls of some special type B semigroups. These results strengthen the results of Fountain and Lawson (Semigroup Forum 32:79–86, 1985) on adequate semigroups. Finally, we give a new proof of a problem posted by Petrich on translational hulls of inverse semigroups in Petrich (Inverse Semigroups, Wiley, New York, 1984).     

Ring semigroups whose subsemigroups form a chain

A multiplicative semigroup S is called a ring semigroup if an addition may be defined on S so that (S,+,⋅) is a ring. Such semigroups have been well-studied in the literature (see Bell in Words, Languages and Combinatorics, pp. 24–31, World Scientific, Singapore, 1994; Jones in Semigroup Forum 47(1):1–6, 1993; Jones and Ligh in Semigroup Forum 17(2):163–173, 1979). In this note, we use Mihăilescu’s Theorem (formerly Catalan’s Conjecture) to characterize the ring semigroups whose subsemigroups containing 0 form a chain with respect to set inclusion.     

The structure of bisimple left unipotent semigroups as ordered pairs

Call a semigroup S left unipotent if eachℒ-class of S contains exactly one idempotent. A structure theorem for bisimple left unipotent semigroups is given which reduces to that of N. R. Reilly [8] for bisimple inverse semigroups. A structure theorem, alternative to one given by R. J. Warne [13], is given for the case when the band E_S of idempotents of S is an ω-chain of right zero semigroups, and two applications of it are made. This research was partially supported by a grant from the National Science Foundation.     

Categorical semigroups

This investigation was stimulated by a question raised by F.R. McMorris and M. Satyanarayana [Proc. Amer. Math. Soc. 33 (1972), 271–277] which asked whether a regular semigroup with a tree of idempotents is categorical. The question is answered in the affirmative. Characterizations of categorical semigroups are found within the following classes of semigroups: regular semigroups, bands, commutative regular semigroups, unions of simple semigroups, semilattices of groups, and commutative semigroups. Some results are related to part of the work of M. Petrich [Trans. Amer. Math. Soc. 170 (1972), 245–268]. For instance, it is shown that the poset of J-classes of any regular categorical semigroup is a tree; however, an example of a regular non-categorical semigroup is given in which the poset of J-classes is a chain. It is also shown that the condition that the subsemigroup of idempotents be categorical is sufficient, but not necessary, for an orthodox semigroup to be categorical.     

Lord Kelvin’s method of images in semigroup theory

We show that Lord Kelvin’s method of images is a way to prove generation theorems for semigroups of operators. To this end we exhibit three examples: a more direct semigroup-theoretic treatment of abstract delay differential equations, a new derivation of the form of the McKendrick semigroup, and a generation theorem for a semigroup describing kinase activity in the recent model of Kaźmierczak and Lipniacki (J. Theor. Biol. 259:291–296, 2009).     

Irregular abelian semigroups with weakly amenable semigroup algebra

In this paper we give counterexamples for the open problem, posed by Blackmore (Semigroup Forum 55:359–377, 1987) of whether weak amenability of a semigroup algebra ℓ ¹(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but ℓ ¹(S) is weakly amenable. Examples are then given.     

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)     

On q *-bisimple IC semigroups of type-E

A semigroup is called type-E if the band of its idempotents can be expressed as a direct product of a rectangular band and an ω-chain. For brevity, we call an IC ^*-bisimple quasi-adequate semigroup of type-E a q ^*-bisimple IC semigroup of type-E. In this paper, we characterize q ^*-bisimple semigroups by using some kind of generalized Bruck-Reilly extensions. As a consequence, some results concerning *-bisimple type-A ω-semigroups given by Asibong-Ibe (Semigroup Forum 31:99–117, 1985) are generalized.     

Tight inverse semigroups that are not bisimple

A semigroup is tight if each of its congruences is uniquely determined by each of the congruence classes. Bisimple inverse semigroups are tight, and tight semigroups are either simple or congruence-free with zero. Although congruence-free semigroups are tight, they are not necessarily bisimple. We construct tight inverse semigroups and tight inverse monoids that are neither bisimple nor congruence-free.     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

Semigroups for flows in infinite networks

Inspired by previous work of M. Kramar and E. Sikolya (Math. Z. 249, 139–162, [2005]), we study transport processes on infinite networks. These “flows” can be modeled by operator semigroups on a suitable Banach space. Using functional analytical and graph theoretical methods, we investigate its spectral properties to determine the long time behavior of the system, and finally characterize uniform convergence of the semigroup to a periodic group under appropriate assumptions on the network.     

Mayer’s transfer operator and representations of GL<Subscript>2</Subscript>

In this paper we relate Mayer’s transfer operator for the geodesic flow of the modular surface to the representation theory of the semigroup of invertible 2×2-matrices with non-negative entries. It turns out that similarly to the case of the Kac-Baker model (see Hilgert et al., Convex Cones, and Semigroups, Oxford University Press, London, 1989 and Hilgert and Mayer, Commun. Math. Phys. 232:19–58, 2002) from statistical mechanics which is related to Howe’s oscillator semigroup one has to introduce an additional multiplication operator to obtain a self-adjoint Hilbert space operator of trace class with the correct spectrum from the natural operators provided by the representation theory. In the present case the representations naturally live on weighted Bergman spaces, but can also be realized on weighted L ²-spaces. Using the representation theory of Ol’shanskiĭ semigroups the semigroup representations can be analytically extended to the simply connected covering of SL(2,ℝ) where they can be identified as holomorphic discrete series representations. To Karl Heinrich Hofmann on the occasion of his 75th birthday.     

Zappa–Szép products of bands and groups

Zappa–Szép products arise when an algebraic structure has the property that every element has a unique decomposition as a product of elements from two given substructures. They may also be constructed from actions of two structures on one another, satisfying axioms first formulated by G. Zappa, and have a natural interpretation within automata theory. We study Zappa–Szép products arising from actions of a group and a band, and study the structure of the semigroup that results. When the band is a semilattice, the Zappa–Szép product is orthodox and ℒ-unipotent. We relate the construction (via automata theory) to the λ-semidirect product of inverse semigroups devised by Billhardt.     

Subalgebras of Orthomodular Lattices

Sachs (Can J Math 14:451–460, 1962) showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham (Int J Theor Phys 37(11):2669–2733, 1998, Int J Theor Phys 38(3):827–859, 1999), at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to this line of work.     

Generalized Cayley graphs of semigroups II

Following Zhu (Semigroup Forum, 2011, doi:), we study generalized Cayley graphs of semigroups. The Cayley D-saturated property, a particular combinatorial property, of generalized Cayley graphs of semigroups is considered and most of the results in Kelarev and Quinn (Semigroup Forum 66:89–96, 2003), Yang and Gao (Semigroup Forum 80:174–180, 2010) are extended. In addition, for some basic graphs and their complete fission graphs, we describe all semigroups whose universal Cayley graphs are isomorphic to these graphs.