A Study of Completely Inverse Paramedial AG-Groupoids

A magma S that meets the identity, xy·z = zy·x, ∀x, y, z ∈ S is called an AG-groupoid. An AG-groupoid S gratifying the paramedial law: uv · wx = xv · wu, ∀ u, v, w, x ∈ S is called a paramedial AGgroupoid. Every AG-grouoid with a left identity is paramedial. We extend the concept of inverse AG-groupoid [4, 7] to paramedial AG-groupoid and investigate various of its properties. We prove that inverses of elements in an inverse paramedial AG-groupoid are unique. Further, we initiate and investigate the notions of congruences, partial order and compatible partial orders for inverse paramedial AG-groupoid and strengthen this idea further to a completely inverse paramedial AG-groupoid. Furthermore, we introduce and characterize some congruences on completely inverse paramedial AG-groupoids and introduce and characterize the concept of separative and completely separative ordered, normal sub-groupoid, pseudo normal congruence pair, and normal congruence pair for the class of completely inverse paramedial AG-groupoids. We also provide a variety of examples and counterexamples for justification of the produced results.

Good congruences on weakly ample semigroups

The concept of normal congruence on a weakly ample semigroup S is introduced and the maximum and minimum admissible congruences whose trace is the normal congruence on a weakly ample semigroup S are characterized in this paper. Some results about congruences on ample semigroups are generalized to weakly ample semigroups.

Surgical Treatment for Malunion of the Lateral Humeral Epicondyle with Posterior Subluxation of the Radial Head: A Case Report and Literature Review

A 24-year-old right-handed man suffered right olecranon and lateral epicondylar fracture from high energy trauma. Fixation of olecranon was performed by a previous doctor. Three months after operation, he presented with limited range of motion (ROM) of the right elbow caused by malunion of the lateral epicondylar fracture and subluxation of the radiohumeral joint. Preoperative ROM of the right elbow was flexion 110° and extension −75°. Forearm rotation was pronation 85° and supination 65°. Fragment excision of the lateral epicondyle, which was 27 mm in length, and lateral collateral ligament repair using anchors were performed. Fourteen months postoperatively, contracture release of the elbow was performed. Twenty-four months postoperatively, radiograph of the elbow showed normal congruence without osteoarthritic changes and the ROM of the right elbow was flexion 120° and extension −35°. Forearm rotation was pronation 90° and supination 70°. In the surgical setting, in case of the size of the lateral epicondylar fragment is relatively large, the fragment should be fixed or lateral collateral ligament should be repaired when the instability of the elbow is found.

Normal Congruence Subgroups of Hecke Groups

Hecke groups are an important class of discrete subgroups of PSL(2, ℝ), which play an important role in the study of Dirichlet series. Subgroups with finite index of a Hecke group, which are called congruence subgroups, are often used. Let q be a positive integer with [Formula: see text]. For the Hecke group [Formula: see text], the structures of principal congruence subgroups and normal congruence subgroups of level m are investigated in many papers, where m is a prime or a power of an odd prime. In this paper, we deal with the case that the level m is a power of 2.

R-Unipotent Congruences on Eventually Regular Semigroups

A semigroup S is called an eventually regular semigroup if for every a ∈ S, there exists a positive integer n such that an is regular. In this paper, the R-unipotent, inverse semigroup and group congruences on an eventually regular semigroup S are described by means of certain congruence pairs (ξ, K), where ξ is a normal congruence on the subsemigroup 〈E(S)〉 generated by E(S), and K is a normal subsemigroup of S.