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.     

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.     

On some classes of regular semigroups

An idempotent e of a semigroup S is called right [left] principal (B.R. Srinivasan, [2]) if fef=fe [fef=ef] for every idempotent f of S. Say that S has property (LR) [(LR₁)] if every ℒ-class of S contains atleast [exactly] one right principal idempotent. There and six further properties obtained by replacing, ‘ℒ-class’ by ‘ℛ-class’ and/or ‘right principal’ by ‘left principal’ are examined. If S has (LR₁), the set of right principal elementsa of S (aa′ is right principal for some inversea′ ofa) is an inverse subsemigroup of S, generalizing a theorem of Srinivasan [2] for weakly inverse semigroups. It is shown that the direct sum of all dual Schützenberger representations of an (LR) semigroup is faithful (cf[1], Theorem 3.21, p. 119). Finally, necessary and sufficient conditions are given on a regular subsemigroup S of the full transformation semigroup on a set in order that S has each of the properties (LR), (LR₁), etc.     

The structure of ?*-inverse semigroups

The concepts of ℒ*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the ℒ*-inverse semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the ℒ*-inverse semigroups by using the left wreath products.     

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.     

On U-ample ω-semigroups

The investigation of U-ample ω-semigroups is initiated. After obtaining some properties of such semigroups, a structure of U-ample ω-semigroups is established. It is proved that a semigroup is a U-ample ω-semigroup if and only if it can be expressed by WBR（T, 0）, namely, the weakly Bruck-Reilly extensions of a monoid T. This result not only extends and amplifies the structure theorem of bisimple inverse ω-semigroups given by N. R. Reilly, but also generalizes the structure theorem of ,-bisimple type A ω-semigroups given by U. Asibong-Ibe in 1985.     

On semigroups in which every Rees one-sided congruence is a congruence

The purpose of this paper is to examine the structure of those semigroups which satisfy one or both of the following conditions: A_r(A_ℓ): The Rees right (left) congruence associated with any right (left) ideal is a congruence. The conditions A_r and A_ℓ are generalizations of commutativity for semigroups. This paper is a continuation of the work of Oehmke [5] and Jordan [4] on H-semigroups (H for hamiltonian, a semigroup is called an H-semigroup if every one-sided congruence is a two-sided congruence). In fact the results of section 2 of Oehmke [5] are proved here under the condition A_r and/or A_ℓ and not the stronger hamiltonian condition. Section 1 of this paper is essentially a summary of the known results of Oehmke. In section 2 we examine the structure of irreducible semigroups satisfying the condition A_r and/or A_ℓ. In particular we determine all regular (torsion) irreducible semigroups satisfying both the conditions A_r and A_ℓ. This research has been supported by Grant A7877 of the National Research Council of Canada.     

半群上Rees矩阵半群的半格的结构

推广了Ｍ．Ｐｅｔｒｉｃｈ在文［１］中所用的方法，得到了幺半群上Ｒｅｅｓ矩阵半群的半格的一个结构定理．研究了单幂幺半群上Ｒｅｅｓ矩阵半群的半格的性质并给出了矩形单幂幺半群的半格的若干等价刻划．     

E-free objects and e-locality for completely regular semigroups

We prove that the e-variety CR(H), of all completely regular semigroups whose subgroups belong to some group variety H, is e-local; that is, every regular, locally completely regular semigroupoid [with subgroups fromH] divides a completely regular semigroup [with subgroups from H], in a ‘regular’ way. In a future paper with P.G. Trotter, this theorem will be applied to semidirect products of e-varieties and to e-free E-solid regular semigroups. A key role in the proof is played by the e-free semigroups in the e-variety CR(H). We provide a solution to the ‘word problem’ in these semigroups, in the style of that for free completely regular semigroups given by Kadourek and Polàk. The solution is derived from the author's work on free products of completely regular semigroups. Communicated by F. Pastijn The author is indebted to the Australian Research Council and to National Science Foundation grant INT-8913404 for their support of this research.     

Finite Proper Covers in a Class of Finite Semigroups with Commuting Idempotents

   Abstract. Weakly left ample semigroups are a class of semigroups that are (2,1) -subalgebras of semigroups of partial transformations, where the unary operation takes a transformation α to the identity map in the domain of α . It is known that there is a class of proper weakly left ample semigroups whose structure is determined by unipotent monoids acting on semilattices or categories. In this paper we show that for every finite weakly left ample semigroup S , there is a finite proper weakly left ample semigroup

On quasi-prime,weakly quasi-prime left ideals in ordered-Γ-semigroups

In this paper we obtain and establish some important results in ordered Γ-semigroups extending and generalizing those for semigroups given in [PETRICH, M.: Introduction to Semigroups, Merill, Columbus, 1973] and for ordered semigroups from [KEHAYOPULU, N.: On weakly prime ideals of ordered semigroups, Math. Japon. 35 (1990), 1051–1056], [KEHAYOPULU, N.: On prime, weakly prime ideals in ordered semigroups, Semigroup Forum 44 (1992), 341–346] and [XIE, X. Y.—WU, M. F.: On quasi-prime, weakly quasi-prime left ideals in ordered semigroups, PU.M.A. 6 (1995), 105–120]. We introduce and give some characterizations about the quasi-prime and weakly quasi-prime left ideals of ordered-Γ-semigroups. We also introduce the concept of weakly m-systems in ordered Γ-semigroups and give some characterizations of the quasi-prime and weakly quasi-prime left ideals by weakly m-systems.     

A generalization of the rees theorem to a class of regular semigroups

Let S be a regular semigroup for which Green's relations J and D coincide, and which is max-principal in the sense that every element of S is contained in maximal principal right, left and two-sided ideals of S. A construction is given of a max-principal regular semigroup W with J=D, which is also principally separated in the sense that distinct maximal principal right (or left) ideals of S are disjoint, and an epimorphism ψ: W→S that preserves maximality of principal left, right, and two sided ideals, and is in a sense locally one-to-one. If S is completely simple, this construction reduces to the Rees matrix representation of S. The main result of this paper has its origin in an incorrect result contained in the author's doctoral dissertation which was written at the University of California (Berkeley) under Professor John Rhodes. This theorem was first established for finite regular semigroups in [1] (Corollary 2.3), and the present generalization of this result to infinite semigroups was suggested by Professor A. H. Clifford, who the author would like to thank for this as well as his generous encouragement and many helpful editorial suggestions. The author would also like to thank Professor Rhodes for his encouragement.     

O-simple strictly regular semigroups

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.     

Semigroups with magnifiers admitting minimal subsemigroups

An element a of a semigroup S is a left magnifier if λ_a, the inner left translation associated with a, is surjective and is not injective (E. S. Ljapin [11]). When this happens there exists a proper subset M of S such that the restriction to M of λ_a is bijective. In that case M is said to be a minimal subset for the left magnifier a (F. Migliorini [13], [14], [15]). Remark that if S is a semigroup having left identities then every left magnifier of S admits minimal subsets which are right ideals. Characterisations for semigroups with left magnifiers which also contain left identities have been given by E. S. Ljapin and R. Desq, using the bicyclic monoid. The general problem, precisely to give a characterization of semigroups having left magnifiers, is still open.     

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.     

Graph expansions of unipotent monoids

Margolis and Meakin use the Cayley graph of a group presentation to construct E-unitary inverse monoids [11]. This is the technique we refer to as graph expansion. In this paper we consider graph expansions of unipotent monoids, where a monoid is unipotent if it contains a unique idempotent. The monoids arising in this way are E-unitary and belong to the quasivariety of weakly left ample monoids. We give a number of examples of such monoids. We show that the least unipotent congruence on a weakly left ample monoid is given by the same formula as that for the least group congruence on an inverse monoid and we investigate the notion of proper for weakly left ample monoids.

Using graph expansions we construct a functor F^e from the category U of unipotent monoids to the category PWLA of proper weakly left ample monoids. The functor F^e is an expansion in the sense of Birget and Rhodes [2]. If we equip proper weakly left ample monoids with an extra unary operation and denote the corresponding category by PWLA ⁰ then regarded as a functor U→PWLA ⁰ F^e is a left adjoint of the functor F^σ : PWLA ⁰ → U that takes a proper weakly left ample monoid to its greatest unipotent image.

Our main result uses the covering theorem of [8] to construct free weakly left ample monoids.