A semigroup with a left identity and left inverse is a group

2021 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Left Identity ◽  
Left Inverse ◽  
Readable Form

We translate the proof of the theorem stated in the title, accomplished by Prover9, into a human readable form.

scholarly journals The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 268 ◽  
Author(s):  
Xiaoying Wu ◽  
Xiaohong Zhang
Keyword(s):  
Disjoint Union ◽  
Decomposition Theorem ◽  
Left Identity ◽  
Maximal Subgroups ◽  
Left Inverse ◽  
Decomposition Theorems

In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the Abel-Grassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups.

scholarly journals Subject Access to a Data Base of Library Holdings

1974 ◽  
Vol 7 (4) ◽  
pp. 267
Author(s):  
Alice S. Clark
Keyword(s):  
Public Service ◽  
Public Libraries ◽  
Ohio State University ◽  
State University ◽  
Complete Collection ◽  
Remote Locations ◽  
Readable Form ◽  
The Ohio State University ◽  
Machine Readable ◽  
Machine Readable Form

As more academic and public libraries have some form of bibliographic description of their complete collection available in machine-readable form, public service librarians are devising ways to use the information for better retrieval. Research at the Ohio State University tested user response to paper and COM output from selected areas of the shelflist. Results indicated users at remote locations found such lists helpful, with some indication that paper printout was more popular than microfiche.

Skew-commuting and Commuting Mappings in Rings with Left Identity

2004 ◽  
Vol 46 (1-2) ◽  
pp. 123-129 ◽  
Author(s):  
R. K. Sharma ◽  
Basudeb Dhara
Keyword(s):  
Left Identity ◽  
Commuting Mappings

scholarly journals Finite AG-groupoid with left identity and left zero

2001 ◽  
Vol 27 (6) ◽  
pp. 387-389 ◽  
Author(s):  
Qaiser Mushtaq ◽  
M. S. Kamran
Keyword(s):  
Commutative Semigroup ◽  
Algebraic Structure ◽  
Zero Element ◽  
Commutative Group ◽  
Left Identity ◽  
Left Zero ◽  
Left Invertive Law

A groupoidGwhose elements satisfy the left invertive law:(ab)c=(cb)ais known as Abel-Grassman's groupoid (AG-groupoid). It is a nonassociative algebraic structure midway between a groupoid and a commutative semigroup. In this note, we show that ifGis a finite AG-groupoid with a left zero then, under certain conditions,Gwithout the left zero element is a commutative group.

Skin Manifestations of Internal Disorders (Dermadromes)

PEDIATRICS ◽  
1948 ◽  
Vol 2 (6) ◽  
pp. 750-750
Author(s):  
P. A. DI SANT' AGNESE
Keyword(s):  
Rheumatic Fever ◽  
Erythema Multiforme ◽  
Scarlet Fever ◽  
Recent Attempt ◽  
Systematic Description ◽  
Skin Manifestations ◽  
Readable Form ◽  
Skin Nodules

This book represents the most recent attempt to provide a systematic description of the skin manifestations of internal disorders. The author has coined and used throughout the volume the term "dermadrome" to designate the dermal component of a syndrome. Thus for instance the characteristic exanthem is the "dermadrome" of scarlet fever, erythema multiforme and skin nodules the "dermadromes" of rheumatic fever. The dermatologic symptoms are detailed under the heading of the various diseases. This facilitates reference within the text for those not conversant with all the intricacies of dermatologic classification. Author and publisher have done a commendable job in presenting the material in clear and readable form.

scholarly journals A Study of Completely Inverse Paramedial AG-Groupoids

2021 ◽  
pp. 101-115
Author(s):  
Muhammad Rashad ◽  
Imtiaz Ahmad ◽  
Faruk Karaaslan
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Left Identity ◽  
Normal Congruence ◽  
Paramedial Law ◽  
Congruence Pair

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.

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