Nil extensions of simple regular ordered semigroups

In this paper, nil extensions of simple regular ordered semigroups, left simple and right regular ordered semigroups, etc. have been characterized. Also, we describe the ordered semigroups which are complete semilattices of nil extensions of left simple and right regular ordered semigroups, left group like ordered semigroups, etc.

Green's Relations for Hypergroupoids

We give some information concerning the Green's relations $\cal R$ and $\cal L$ in hypergroupoids extending the concepts of right (left) consistent or intra-consistent groupoids in case of hypergroupoids. We prove, for example, that if an hypergroupoid $H$ is right (left) consistent or intra-consistent, then the Green's relations $\cal R$ and $\cal L$ are equivalence relations on $H$ and give some conditions under which in consistent commutative hypergroupoids the relation $\cal R$ (= $\cal L$) is a semilattice congruence. A commutative hypergroupoid is right consistent if and only if it is left consistent and if an hypergroupoid is commutative and right (left) consistent, then it is intra-consistent. A characterization of right (left) consistent (or intra-consistent) right (left) simple hypergroupoids has been also given. Illustrative examples are given.

On left regular and intra-regular ordered semigroups

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.

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.

Some properties of hyperideals in ternary semihypergroups

Ternary semihypergroups are algebraic structures with one ternary associative hyperoperation. In this paper we give some properties of left (right) and lateral hyperideals in ternary semihypergroups. We introduce the notion of left simple, lateral simple, left (0-)simple and lateral 0-simple ternary semihypergroups and characterize the minimality and maximality of left (right) and lateral hyperideals in ternary semihypergroups. The relationship between them is investigated in ternary semihypergroups extending and generalizing the analogues results for ternary semigroups.

Strategy for Treating Simple Testicular Cyst in Adults

Simple testicular cyst is increasingly diagnosed because of the general availability of high-resolution ultrasound devices, although its management has been controversial. The authors report their experiences in managing large simple testicular cyst in two adults aged 56 and 68 years. The first patient was hospitalized with a presumptive diagnosis of left hydrocele, and the second patient was hospitalized with left simple testicular cyst. Both patients felt pain in the left scrotum. The maximum diameter of the cyst in both patients was more than 6 cm. The patients were successfully treated with orchiectomy with very good long-term results. The authors indicate that a careful study of the medical history, physical examination, and scrotal ultrasonography may facilitate an accurate preoperative diagnosis. Age, symptom, compliance with the surveillance of the patient, and the size and dynamic ultrasonographic changes of the cyst should all be considered in the selection of the treatment regimen. Such a study may be helpful in managing a simple testicular cyst.

𝒩-Fuzzy Ideals in Ordered Semigroups

We introduce the concept of𝒩-fuzzy left (right) ideals in ordered semigroups and characterize ordered semigroups in terms of𝒩-fuzzy left (right) ideals. We characterize left regular (right regular) and left simple (right simple) ordered semigroups in terms of𝒩-fuzzy left (𝒩-fuzzy right) ideals. The semilattice of left (right) simple semigroups in terms of𝒩-fuzzy left (right) ideals is discussed.