Open AccessA Study of Completely Inverse Paramedial AG-Groupoids
Author(s) -
Muhammad Rashad,
Imtiaz Ahmad,
Faruk Karaaslan
Publication year - 2021
Publication title -
the punjab university journal of mathematics
Language(s) - English
Resource type - Journals
ISSN - 1016-2526
DOI - 10.52280/pujm.2021.530202
Subject(s) - congruence relation , mathematics , inverse , double groupoid , pure mathematics , counterexample , congruence (geometry) , identity (music) , class (philosophy) , inverse semigroup , combinatorics , algebra over a field , computer science , geometry , physics , artificial intelligence , acoustics
A magma S that meets the identity, xy·z = zy·x, ∀x, y, z ∈ Sis called an AG-groupoid. An AG-groupoid S gratifying the paramediallaw: uv · wx = xv · wu, ∀ u, v, w, x ∈ S is called a paramedial AGgroupoid. Every AG-grouoid with a left identity is paramedial. We extendthe concept of inverse AG-groupoid [4, 7] to paramedial AG-groupoid andinvestigate various of its properties. We prove that inverses of elements inan inverse paramedial AG-groupoid are unique. Further, we initiate andinvestigate the notions of congruences, partial order and compatible partialorders for inverse paramedial AG-groupoid and strengthen this idea further to a completely inverse paramedial AG-groupoid. Furthermore, weintroduce 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, pseudonormal congruence pair, and normal congruence pair for the class of completely inverse paramedial AG-groupoids. We also provide a variety ofexamples and counterexamples for justification of the produced results.