A Class of Homomorphism Theories for Groupoids

In a well-behaved homomorphism theory for a class of algebraic systems certain “closed objects” relative to a given G ∈ are distinguished which act as kernels of homomorphisms. For example, if is the class of groups then the closed objects relative to a given group G are the normal subgroups of G; if is the class of semigroups with zero element then one can devise a homomorphism theory in which the closed objects relative to a given S ∈ are the ideals of S[cf. Rees (3)]; in the class of groupoids one may define the closed objects relative to a given groupoid G to be the congruence relations on G, that is, subsets π⊆G×G which are equivalence relations having the property that (x1y1, x2y2) ∈ π whenever (x1, x2), (y1, y2) ∈ π. Given such a closed object N relative to G there exists a “factor” system G/N and a (canonical) homomorphism η: G→G/N characterised by the property: If σ G→H is a homomorphism with kernel N then there is a unique homomorphism : G/N→H such that . η = σ and the kernel of is trivial in the sense that the kernel of is the unique smallest closed object relative to G/N.

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Equivalence Relations in Algebraic Systems

American Mathematical Monthly ◽

10.1080/00029890.1949.11999401 ◽

1949 ◽

Vol 56 (6) ◽

pp. 372-377

Author(s):

R. R. Stoll

Keyword(s):

Equivalence Relations ◽

Algebraic Systems

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Least fixpoints of endofunctors of cartesian closed categories

Mathematical Structures in Computer Science ◽

10.1017/s0960129500000190 ◽

1993 ◽

Vol 3 (2) ◽

pp. 229-257 ◽

Cited By ~ 5

Author(s):

J. Lambek

Keyword(s):

Propositional Calculus ◽

Second Order ◽

Equivalence Relations ◽

Cartesian Closed Category ◽

Closed Category ◽

Congruence Relations ◽

Partial Recursive Functions ◽

Cartesian Closed Categories ◽

Partial Equivalence Relations ◽

Cartesian Closed

Least fixpoints are constructed for finite coproducts of definable endofunctors of Cartesian closed categories that have weak polynomial products and joint equalizers of arbitrary families of pairs of parallel arrows. Both conditions hold in PER, the category whose objects are partial equivalence relations on N, and whose arrows are partial recursive functions. Weak polynomial products exist in any cartesian closed category with a finite number of objects as well as in any model of second order polymorphic lambda calculus: that is, in the proof theory of any second order positive intuitionistic propositional calculus, but such a category need not have equalizers. However, any finite coproduct of definable endofunctors of a cartesian closed category with weak polynomial products will have a least fixpoint in a larger category with equalizers whose objects are right ideals (or sieves) of modulo certain congruence relations, and whose arrows are induced from .

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Maximal n-generated subdirect products

International Journal of Algebra and Computation ◽

10.1142/s0218196716500065 ◽

2016 ◽

Vol 26 (01) ◽

pp. 123-155

Author(s):

Joel Berman

Keyword(s):

Upper Bound ◽

Finite Algebra ◽

Equivalence Relations ◽

Subdirectly Irreducible ◽

Finite Algebras ◽

Locally Finite Variety ◽

Congruence Relations ◽

Finite Set ◽

Basic Parameters ◽

Subdirectly Irreducible Algebras

For [Formula: see text] a positive integer and [Formula: see text] a finite set of finite algebras, let [Formula: see text] denote the largest [Formula: see text]-generated subdirect product whose subdirect factors are algebras in [Formula: see text]. When [Formula: see text] is the set of all [Formula: see text]-generated subdirectly irreducible algebras in a locally finite variety [Formula: see text], then [Formula: see text] is the free algebra [Formula: see text] on [Formula: see text] free generators for [Formula: see text]. For a finite algebra [Formula: see text] the algebra [Formula: see text] is the largest [Formula: see text]-generated subdirect power of [Formula: see text]. For every [Formula: see text] and finite [Formula: see text] we provide an upper bound on the cardinality of [Formula: see text]. This upper bound depends only on [Formula: see text] and these basic parameters: the cardinality of the automorphism group of [Formula: see text], the cardinalities of the subalgebras of [Formula: see text], and the cardinalities of the equivalence classes of certain equivalence relations arising from congruence relations of [Formula: see text]. Using this upper bound on [Formula: see text]-generated subdirect powers of [Formula: see text], as [Formula: see text] ranges over the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text], we obtain an upper bound on [Formula: see text]. And if all the [Formula: see text]-generated subdirectly irreducible algebras in [Formula: see text] have congruence lattices that are chains, then we characterize in several ways those [Formula: see text] for which this upper bound is obtained.

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Boolean Zero Square (BZS) Semigroups

Sultan Qaboos University Journal for Science [SQUJS] ◽

10.53539/squjs.vol26iss1pp31-39 ◽

2021 ◽

Vol 26 (1) ◽

pp. 31-39

Author(s):

Pinto G.A.

Keyword(s):

Inverse Semigroup ◽

Simple Semigroup ◽

Sufficient Conditions ◽

Zero Element ◽

Necessary And Sufficient Conditions ◽

Equivalence Relations ◽

New Class ◽

Necessary And Sufficient

We introduce a new class of semigroups, that we call BZS - Boolean Zero Square-semigroups. A semigroup S with a zero element, 0, is said to be a BZS semigroup if, for every , we have or . We obtain some properties that describe the behaviour of the Green’s equivalence relations , , and . Necessary and sufficient conditions for a BZS semigroup to be a band and an inverse semigroup are obtained. A characterisation of a special type of BZS completely 0-simple semigroup is presented.

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On normal subgroups of direct products

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500018228 ◽

1990 ◽

Vol 33 (2) ◽

pp. 309-319 ◽

Cited By ~ 3

Author(s):

F. E. A. Johnson

Keyword(s):

Automorphism Group ◽

Representation Theory ◽

Finite Number ◽

Equivalence Classes ◽

Equivalence Relations ◽

Direct Products ◽

Normal Subgroups ◽

Free Abelian Groups ◽

Subdirect Products ◽

Number Of Classes

We investigate the equivalence classes of normal subdirect products of a product of free groups Fn1 × … × Fnk under the simultaneous equivalence relations of commensurability and conjugacy under the full automorphism group. By abelianisation, the problem is reduced to one in the representation theory of quivers of free abelian groups. We show there are infinitely many such classes when k≧3, and list the finite number of classes when k = 2.

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Norms over intuitionistic fuzzy congruence relations on rings

Notes on Intuitionistic Fuzzy Sets ◽

10.7546/nifs.2021.27.3.51-68 ◽

2021 ◽

Vol 27 (3) ◽

pp. 51-68

Author(s):

Rasul Rasuli ◽

Keyword(s):

Equivalence Relation ◽

Congruence Relation ◽

Equivalence Relations ◽

Intuitionistic Fuzzy ◽

Congruence Relations ◽

Fuzzy Congruence

In this paper, by using norms, we define the concept of intuitionistic fuzzy equivalence relations and intuitionistic fuzzy congruence relations on ring R and we investigate some assertions. Also we define intuitionistic fuzzy ideals of ring R under norms and compare this with fuzzy equivalence relation and fuzzy congruence relation on ring R such that we define new introduced ring.

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Soft relations applied to semigroups

Filomat ◽

10.2298/fil1307183f ◽

2013 ◽

Vol 27 (7) ◽

pp. 1183-1196 ◽

Cited By ~ 17

Author(s):

Feng Feng ◽

Irfan Ali ◽

Muhammad Shabir

Keyword(s):

Binary Relation ◽

Semigroup Theory ◽

Fuzzy Relation ◽

Equivalence Relations ◽

Mathematical Framework ◽

Binary Relations ◽

Soft Sets ◽

Fixed Parameter ◽

Congruence Relations ◽

Isomorphism Theorems

Binary relations, in particular, equivalence relations play an important role in both mathematics and information sciences. The concept of soft sets was initiated by Molodtsov as a general mathematical framework for dealing with uncertainty. The present paper establishes a possible connection between binary relations and soft sets. The concept of soft binary relations is introduced and some related properties are investigated. It is shown that any fuzzy relation may be considered as a soft binary relation. Moreover, we discuss the application of soft binary relations in semigroup theory. We consider soft congruence relations over semigroups and show that all soft congruence relations over a semigroup with a fixed parameter set form a lattice. Finally, the notion of soft homomorphisms is presented and isomorphism theorems for soft semigroups are established based on soft congruence relations.

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