EQUATIONS OVER DIRECT POWERS OF ALGEBRAIC STRUCTURES IN RELATIONAL LANGUAGES

For a semigroup S (group G) we study relational equations and describe all semigroups S with equationally Noetherian direct powers. It follows that any group G has equationally Noetherian direct powers if we consider G as an algebraic structure of a certain relational language. Further we specify the results as follows: if a direct power of a finite semigroup S is equationally Noetherian, then the minimal ideal Ker(S) of S is a rectangular band of groups and Ker(S) coincides with the set of all reducible elements

A semigroup is a rectangular band if and only if it is nowhere commutative

We prove the proposition addressed in the title of this paper.

The product of quasi-ideal refined generalised quasi-adequate transversals

As the real common generalisations of both orthodox transversals and adequate transversals in the abundant case, the concept of refined generalised quasi-adequate transversal, for short, RGQA transversal was introduced by Kong and Wang. In this paper, an interesting characterization for a generalised quasi-adequate transversal to be refined is acquired. It is shown that the product of every two quasi-ideal RGQA transversals of the abundant semigroup S satisfying the regularity condition is a quasi-ideal RGQA transversal of S and that all quasi-ideal RGQA transversals of S compose a rectangular band. The related results concerning adequate transversals are generalised and enriched.

MATRIX REPRESENTATION OF RECTANGULAR BANDS OF SEMIGROUPS

Abundant semigroups expressed as a perfect rectangular band of adequate semigroups will be investigated. A known theorem of Pastijn–Petrich in 1984 on perfect rectangular bands of inverse semigroups will be generalized from the class of regular semigroups to the class of abundant semigroups. A Rees matrix representation of such abundant semigroups will be established and some characterization theorems will be consequently given.

Morphological and molecular studies on life cycle stages of Diphtherostomum brusinae (Digenea: Zoogonidae) from northern Portugal

Diphtherostomum brusinae was first recorded by the present study in the north of Portugal. Sporocysts, containing cercariae and encysted metacercariae, were observed in the gonads and digestive gland of the gastropod Nassarius reticulatus. Metacercariae were also found infecting the foot, mantle border and gills of the cockle Cerastoderma edule. The adult form was lodged in the rectum of the definitive host Diplodus sargus. The morphology of the three parasitic stages was studied by light (LM) and scanning electron microscopy (SEM). Despite the close similarity between cercaria and metacercaria, SEM data provided information that allowed their differentiation, namely the presence of a dense crown of microvilli around the oral cavity of the cercariae, which was absent in the metacercariae. In addition, the metacercariae presented a specific pre-acetabular rectangular band with conspicuous triangular spines. The adult showed characteristics of D. brusinae species, in particular the presence of acetabular lips, compact vitellaria and large elliptical eggs. Sequenced ITS1 data clearly demonstrated that the cercariae and metacercarial cysts from N. reticulatus, the cysts from C. edule and the adult isolated from D. sargus were life cycle stages that belonged to the same species, i.e. D. brusinae. Two transmission strategies in the life cycle of this species were observed: (1) cercariae encyst within the sporocysts of N. reticulatus and await ingestion by the definitive host; and (2) N. reticulatus naturally emits cercariae; they encyst in C. edule or the environment and are ingested by the definitive host.

A STRUCTURE THEOREM FOR PERFECT ABUNDANT SEMIGROUPS

The direct product of a cancellative monoid and a rectangular band is called a can-cellative plank. In this paper, we describe the semigroups which can be expressed as a strong semilattice of cancellative planks. Our result not only generalizes the well known 1951 Clifford theorem for completely regular semigroups having central idempotents, but also the theorem for C-rpp monoids, that is, left abundant monoids having central idempotents, given by Fountain in 1977. Some recent results of the authors concerning rpp semigroups belonging to a class we call perfect are strengthened.

Certain characterizations of varieties of bands

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

RP-dominated regular semigroups

SynopsisWe describe the structure of regular semigroups in which each element is dominated, in the Nambooripad order, by a unique maximal element that is ℋ-equivalent to a mididentity. These are precisely direct products of a rectangular band and a uniquely unit regular semigroup.