Congruence pairs of principal p-algebras

2017 ◽  
Vol 67 (1) ◽  
pp. 263-270
Author(s):  
Abd El-Mohsen Badawy ◽  
Kar Ping Shum
Keyword(s):  
Congruence Relation ◽  
Congruence Pair

Abstract In this paper, we first introduce the concept of congruence pairs on the class of principal p-algebras. Then, we show that every congruence relation θ on a principal p-algebra L can be uniquely determined by a congruence pair (θ 1, θ 2). Moreover, the strong extensions of principal p-algebras and the permutability of congruences will be investigated by congruence pairs.

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

2020 ◽  
Vol 70 (6) ◽  
pp. 1275-1288
Author(s):  
Abd El-Mohsen Badawy ◽  
Miroslav Haviar ◽  
Miroslav Ploščica
Keyword(s):  
Boolean Algebra ◽  
Congruence Lattices ◽  
Descriptive Solution ◽  
The One ◽  
Special Case ◽  
Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

scholarly journals States and Measures on Hyper BCK-Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xiao-Long Xin ◽  
Pu Wang
Keyword(s):  
Congruence Relation ◽  
Basic Properties ◽  
Mv Algebra ◽  
Regular Congruence ◽  
Bosbach State

We define the notions of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra(H,∘,0,e)and derive some basic properties of them. We construct a quotient hyper BCK-algebra via a regular congruence relation. We also define a∘-compatibledregular congruence relationθand aθ-compatibledinf-Bosbach stateson(H,∘,0,e). By inducing an inf-Bosbach states^on the quotient structureH/[0]θ, we show thatH/[0]θis a bounded commutative BCK-algebra which is categorically equivalent to an MV-algebra. In addition, we introduce the notions of hyper measures (states/measure morphisms/state morphisms) on hyper BCK-algebras, and present a relation between hyper state-morphisms and Bosbach states. Then we construct a quotient hyper BCK-algebraH/Ker(m)by a reflexive hyper BCK-idealKer(m). Further, we prove thatH/Ker(m)is a bounded commutative BCK-algebra.

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.

Fuzzy congruence relation generated by a fuzzy relation in vector spaces

2018 ◽  
Vol 35 (5) ◽  
pp. 5635-5645
Author(s):  
S. Khosravi Shoar ◽  
R.A. Borzooei ◽  
R. Moradian
Keyword(s):  
Congruence Relation ◽  
Fuzzy Relation ◽  
Vector Spaces ◽  
Fuzzy Congruence

scholarly journals Involution Abel–Grassmann’s Groups and Filter Theory of Abel–Grassmann’s Groups

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 553 ◽  
Author(s):  
Xiaohong Zhang ◽  
Xiaoying Wu
Keyword(s):  
Congruence Relation ◽  
Filter Theory ◽  
Basic Properties ◽  
New Concepts ◽  
Structure Theorems

In this paper, some basic properties and structure characterizations of AG-groups are further studied. First, some examples of infinite AG-groups are given, and weak commutative, alternative and quasi-cancellative AG-groups are discussed. Second, two new concepts of involution AG-group and generalized involution AG-group are proposed, the relationships among (generalized) involution AG-groups, commutative groups and AG-groups are investigated, and the structure theorems of (generalized) involution AG-groups are proved. Third, the notion of filter of an AG-group is introduced, the congruence relation is constructed from arbitrary filter, and the corresponding quotient structure and homomorphism theorems are established.

scholarly journals Enumeration of Indices of given Altitude and Potency

1959 ◽  
Vol 11 (4) ◽  
pp. 207-209 ◽  
Author(s):  
H. Minc
Keyword(s):  
Congruence Relation

Indices of the free logarithmetic correspond to bifurcating root-trees (cf.(4)), to Evans' non-associative numbers (3) and to Etherington's partitive numbers (2). The free commutative logarithmetic is the homomorph of f determined by the congruence relation P + Q ∼ Q + P. Formulæ for aδ and pα, i.e. the numbers of indices of of a given potency* δ and the number of indices of a given altitude α respectively, were given by Etherington (1), who also gave corresponding formulæ for commutative indices of . Other enumeration formulæ are contained in (5).

On the Structure of Finitely Presented Lattices

1981 ◽  
Vol 33 (2) ◽  
pp. 404-411 ◽  
Author(s):  
G. Gratzer ◽  
A. P. Huhn ◽  
H. Lakser
Keyword(s):  
Congruence Relation ◽  
Lattice Theory ◽  
Structure Theorem ◽  
Congruence Class ◽  
Free Lattice ◽  
Partial Lattice ◽  
Finitely Presented

A lattice L is finitely presented (or presentable) if and only if it can be described with finitely many generators and finitely many relations. Equivalently, L is the lattice freely generated by a finite partial lattice A, in notation, L = F(A). (For more detail, see Section 1.5 of [6].)It is an old “conjecture” of lattice theory that in a finitely presented (or presentable) lattice the elements behave “freely” once we get far enough from the generators.In this paper we prove a structure theorem that could be said to verify this conjecture.THEOREM 1. Let L be a finitely presentable lattice. Then there exists a congruence relation θ such that L/θ is finite and each congruence class is embeddable in a free lattice.

scholarly journals The Homomorphisms and Operations of Rough Groups

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Fei Li ◽  
Zhenliang Zhang
Keyword(s):  
Normal Subgroup ◽  
Congruence Relation ◽  
Quotient Groups

Based on the light relation between a normal subgroup and a complete congruence relation of a group, we consider the homomorphism problem of rough groups and rough quotient groups and investigate their operational properties. Some new results are obtained.

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