On ordered semigroups which are semilattices of left simple semigroups

It has been proved by Tôru Saitô that a semigroup S is a semilattice of left simple semigroups, that is, it is decomposable into left simple semigroups, if and only if the set of left ideals of S is a semilattice under the multiplication of subsets, and that this is equivalent to say that S is left regular and every left ideal of S is two-sided. Besides, S. Lajos has proved that a semigroup S is left regular and the left ideals of S are two-sided if and only if for any two left ideals L ₁, L ₂ of S, we have L ₁ ∩ L ₂ = L ₁ L ₂. The present paper generalizes these results in case of ordered semigroups. Some additional information concerning the semigroups (without order) are also obtained.     

Orthodox semigroups whose idempotents satisfy a certain identity

An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy [xyx=yx]. Bisimple left [right] inverse semigroups have been studied by Venkatesan [6]. In this paper, we clarify the structure of general left [right] inverse semigroups. Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx=xyzx. In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.     

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.     

Congruences on generalized inverse semigroups

A generalized inverse semigroup is a regular semigroup whose idempotents satisfy a permutation identity X₁ X₂...X_n=X_p1 X_p2...X_pn, where (P₁, P₂..., P_n) is a nontrivial permutation of (1, 2,..., n). Yamada [4] has given a complete classification of generalized inverse semigroups in terms of inverse semigroups, left normal bands, and right normal bands. In this paper we show that every congruence on a generalized inverse semigroup is uniquely determined by a congruence on its associated inverse semigroup, left normal band, and right normal band. A converse is also provided. This paper is extracted from the doctoral thesis of the author written at Monash University under the direction of Professor G. B. Preston. The research was carried out while the author held a Commonwealth Postgraduate Award.     

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.     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

On the Embedding of Ordered Semigroups into Ordered Group

It was shown in [7] that any right reversible, cancellative ordered semigroup can be embedded into an ordered group and as a consequence, it was shown that a commutative ordered semigroup can be embedded into an ordered group if and only if it is cancellative. In this paper we introduce the concept of L-maher and R-maher semigroups and use a technique similar to that used in [7] to show that any left reversible cancellative ordered L or R-maher semigroup can be embedded into an ordered group.     

Cancellative Orders

left order in Q and Q is a semigroup of left quotients of S if every q∈Q can be written as q=a^*b for some a, b∈S where a^* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the {\cal D}-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S. The authors are grateful to the ARC for its generous financial support.     

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.     

WLR-regular orthogroups

A regular orthogroup S with the property that D _e =R _e or D _e =L _e for any idempotent e∈S is called a WLR-regular orthogroup. In this paper, we give constructions of such semigroups in terms of spined products of left and right regular orthogroups with respect to Clifford semigroups. WLR-cryptogroups and its special cases are also investigated. Research supported by General Scientific Research Project of Shanghai Normal University No. SK200707.     

On the Finite and Non-finite Generation of Finitary Power Semigroups

The finitary power semigroup of a semigroup S, denoted P_f(S), is the set of finite subsets of S with respect to the usual set multiplication. Semigroups with finitely generated finitary power semigroups are characterised in terms of three other properties. From this statement there are drawn several corollaries. It follows that P_f(S) is not finitely generated if S is infinite and in any of the following classes: commutative; Bruck-Reilly extensions; inverse semigroups that contain an infinite group; completely zero-simple; completely regular.     

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.     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

Embedding inverse semigroups in coset semigroups

The setK(G) of all cosets X of a group G, modulo all subgroups of G, forms an inverse semigroup under the multiplication X*Y=smallest coset that constains XY. In this note we show that each inverse semigroup S can be embedded in some coset semigroupK(G). This follows from a result which shows that symmetric inverse semigroups can be embedded in the coset semigroups of suitable symmetric groups. We also give necessary and sufficient conditions on an inverse semigroup S in order that it should be isomorphic to someK(G). This research was supported by a grant from the National Science Foundation.     

The structure of L*-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 semigroupΓand a left regular band B together with a mapping which maps the semigroupΓinto 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.     

Fundamental Ehresmann semigroups

\noindent The celebrated construction by Munn of a fundamental inverse semigroup T _E from a semilattice E provides an important tool in the study of inverse semigroups. We present here a semigroup C _E that plays the T _E role for Ehresmann semigroups. Inverse semigroups are Ehresmann, as are those that are adequate, weakly ample or weakly hedged. We describe explicitly the semigroups C _E for some specific semilattices E and extract information relating to the corresponding classes of Ehresmann semigroups. October 13, 1999