IRREDUCIBLE ELEMENTS IN ALGEBRAIC LATTICES

2010 ◽  
Vol 20 (08) ◽  
pp. 969-975 ◽  
Author(s):  
U. M. SWAMY ◽  
B. VENKATESWARLU
Keyword(s):  
Universal Algebra ◽  
Complete Lattice ◽  
Sufficient Conditions ◽  
Algebraic Lattice ◽  
Complete Distributivity ◽  
Lattice Of Subalgebras ◽  
Irreducible Elements ◽  
Algebraic Lattices ◽  
Strongly Irreducible ◽  
Necessary And Sufficient

α-Irreducible and α-Strongly Irreducible Ideals of a ring have been characterized in [2] and [4]. A complete lattice which is generated by compact elements is called an algebraic lattice for the simple reason that every such lattice is isomorphic to the lattice of subalgebras of a suitable universal algebra and vice-versa. In this paper, we characterize the irreducible elements and strongly irreducible elements in an algebraic lattice, which extends the results in [4] to arbitrary algebraic lattices. Also we obtain certain necessary and sufficient conditions, in terms of irreducible elements, for an algebraic lattice to satisfy the complete distributivity.

scholarly journals A natural representation of partitions as terms of a universal algebra

1993 ◽  
Vol 55 (3) ◽  
pp. 311-324
Author(s):  
Harry Lakser
Keyword(s):  
Universal Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Variety Of Algebras ◽  
Natural Representation ◽  
Necessary And Sufficient ◽  
Element Set

AbstractWe consider a variety of algebras with two binary commutative and associative operations. For each integer n ≥ 0, we represent the partitions on an n-element set as n-ary terms in the variety. We determine necessary and sufficient conditions on the variety ensuring that, for each n, these representing terms be all the essentially n-ary terms and moreover that distinct partitions yield distinct terms.

L-fuzzy cosets in universal algebras

2021 ◽  
Vol 71 (3) ◽  
pp. 573-594
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
General Theory ◽  
Universal Algebra ◽  
Fuzzy Set ◽  
Fuzzy Systems ◽  
Algebraic Closure ◽  
Sufficient Conditions ◽  
General Context ◽  
Variety Of Algebras ◽  
Universal Algebras ◽  
Necessary And Sufficient

Abstract In this paper, we introduce the notion of fuzzy costs in a more general context, in universal algebra by the use of coset terms. We study the structure of fuzzy cosets by applying the general theory of algebraic fuzzy systems. Fuzzy cosets generated by a fuzzy set are characterized in different ways. It is also proved that the class of fuzzy cosets determined by an element forms an algebraic closure fuzzy set system. Finally, we give a set of necessary and sufficient conditions for a given variety of algebras to be congruence permutable by applying the theory of fuzzy cosets.

scholarly journals A new view of relationship between atomic posets and complete (algebraic) lattices

2017 ◽  
Vol 15 (1) ◽  
pp. 238-251
Author(s):  
Bin Yu ◽  
Qingguo Li ◽  
Huanrong Wu
Keyword(s):  
Complete Lattice ◽  
Algebraic Lattice ◽  
Complete Lattices ◽  
New Methods ◽  
Algebraic Lattices ◽  
New View

AbstractIn the context of the atomic poset, we propose several new methods of constructing the complete lattice and the algebraic lattice, and the mutual decision of relationship between atomic posets and complete lattices (algebraic lattices) is studied.

scholarly journals A characterization of left semiartinian rings

1974 ◽  
Vol 11 (3) ◽  
pp. 425-428 ◽  
Author(s):  
Jonathan S. Golan
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Krull Dimension ◽  
Semiartinian Ring ◽  
Torsion Theories ◽  
Necessary And Sufficient

In defining the torsion-theoretic Krull dimension of an associative ring R we make use of a function δ from the complete lattice of all subsets of the torsion-theoretic spectrum of R to the complete lattice of all hereditary torsion theories on R-mod. In this note we give necessary and sufficient conditions for δ to be injective, surjective, and bijective. In particular, δ is bijective if and only if R is a left semiartinian ring.

Some properties of retract lattices of monounary algebras

2012 ◽  
Vol 62 (2) ◽  
Author(s):  
Danica Jakubíková-Studenovská ◽  
Jozef Pócs
Keyword(s):  
Sufficient Conditions ◽  
Boolean Lattice ◽  
Necessary And Sufficient Conditions ◽  
Monounary Algebra ◽  
Irreducible Elements ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions for a connected monounary algebra (A, f), under which the lattice R ∅(A, f) of all retracts of (A, f) (together with ∅) is algebraic, are proved. Simultaneously, all connected monounary algebras in which each retract is a union of completely join-irreducible elements of R ∅(A, f) are characterized. Further, there are described all connected monounary algebras (A, f) such that the lattice R ∅(A, f) is complemented. In this case R ∅(A, f) forms a boolean lattice.

m-Algebraic lattices in formal concept analysis

2019 ◽  
Vol 29 (10) ◽  
pp. 1556-1574
Author(s):  
Zhongxi Zhang ◽  
Qingguo Li ◽  
Nan Zhang
Keyword(s):  
Complete Lattice ◽  
Formal Concept Analysis ◽  
Concept Lattice ◽  
Concept Analysis ◽  
Continuous Functions ◽  
Algebraic Lattice ◽  
Formal Concept ◽  
Algebraic Lattices ◽  
Special Cases ◽  
Cartesian Closed

AbstractThe notion of an m-algebraic lattice, where m stands for a cardinal number, includes numerous special cases, such as complete lattice, algebraic lattice, and prime algebraic lattice. In formal concept analysis, one fundamental result states that every concept lattice is complete, and conversely, each complete lattice is isomorphic to a concept lattice. In this paper, we introduce the notion of an m-approximable concept on each context. The m-approximable concept lattice derived from the notion is an m-algebraic lattice, and conversely, every m-algebraic lattice is isomorphic to an m-approximable concept lattice of some context. Morphisms on m-algebraic lattices and those on contexts are provided, called m-continuous functions and m-approximable morphisms, respectively. We establish a categorical equivalence between LATm, the category of m-algebraic lattices and m-continuous functions, and CXTm, the category of contexts and mapproximable morphisms.We prove that LATm is cartesian closed whenevermis regular and m > 2. By the equivalence of LATm and CXTm, we obtain that CXTm is also cartesian closed under same circumstances. The notions of a concept, an approximable concept, and a weak approximable concept are showed to be special cases of that of an m-approximable concept.

scholarly journals Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Berhanu Assaye Alaba ◽  
Wondwosen Zemene Norahun
Keyword(s):  
Complete Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Fuzzy Filter ◽  
Pseudocomplemented Semilattice ◽  
Fuzzy Ideal ◽  
Fuzzy Congruence ◽  
Necessary And Sufficient ◽  
Fuzzy Filters

In this paper, we introduce the concept of kernel fuzzy ideals and ⁎-fuzzy filters of a pseudocomplemented semilattice and investigate some of their properties. We observe that every fuzzy ideal cannot be a kernel of a ⁎-fuzzy congruence and we give necessary and sufficient conditions for a fuzzy ideal to be a kernel of a ⁎-fuzzy congruence. On the other hand, we show that every fuzzy filter is the cokernel of a ⁎-fuzzy congruence. Finally, we prove that the class of ⁎-fuzzy filters forms a complete lattice that is isomorphic to the lattice of kernel fuzzy ideals.

scholarly journals Representation of slim lattice by poset

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 919-925
Author(s):  
Marijana Gorjanac-Ranitovic ◽  
Andreja Tepavcevic
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Slim Lattice ◽  
Irreducible Elements ◽  
Finite Lattices ◽  
Necessary And Sufficient

In this paper, we present the necessary and sufficient conditions for a poset to be a poset of the union of join and meet irreducible elements of the slim lattice. Slim lattices are special finite lattices that are intensively investigated recently. The problem that we solved in this paper is a generalization of the problem proposed very recently by Cz?dli.

scholarly journals On rings of sets

1968 ◽  
Vol 8 (4) ◽  
pp. 723-730 ◽  
Author(s):  
T. P. Speed
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Sets ◽  
The Past ◽  
Complete Ring ◽  
Irreducible Elements ◽  
Necessary And Sufficient

In the past a number of papers have appeared which give representations of abstract lattices as rings of sets of various kinds. We refer particularly to authors who have given necessary and sufficient conditions for an abstract lattice to be lattice isomorphic to a complete ring of sets, to the lattice of all closed sets of a topological space, or to the lattice of all open sets of a topological space. Most papers on these subjects give the conditions in terms of special elements of the lattice. We thus have completely join-irreducible elements — G. N. Raney [7]; join prime, completely join prime, and supercompact elements — V. K. Balachandran [1], [2]; N-sub-irreducible elements — J. R. Büchi [5]; and lattice bisectors — P. D. Finch [6]. Also meet-irreducible and completely meet-irreducible dual ideals play a part in some representations of G. Birkhoff & 0. Frink [4].

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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