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.

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.

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 On congruence lattices of m-complete lattices

1992 ◽  
Vol 52 (1) ◽  
pp. 57-87 ◽  
Author(s):  
G. Grätzer ◽  
H. Lakser
Keyword(s):  
Complete Lattice ◽  
Extension Theorem ◽  
Unit Element ◽  
Algebraic Lattice ◽  
Converse Statement ◽  
Complete Lattices ◽  
Point Extension ◽  
Congruence Lattices ◽  
Congruence Relations ◽  
The One

AbstractThe lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In an earlier paper, we characterize this lattice as a complete lattice. Let m be an uncountable regular cardinal. The lattice L of all m-complete congruence relations of an m-complete lattice K is an m-algebraic lattice; if K is bounded, then the unit element of L is m-compact. Our main result is the converse statement: For an m-algebraic lattice L with an m-compact unit element, we construct a bounded m-complete lattice K such that L is isomorphic to the lattice of m-complete congruence relations of K. In addition, if L has more than one element, then we show how to construct K so that it will also have a prescribed automorphism group. On the way to the main result, we prove a technical theorem, the One Point Extension Theorem, which is also used to provide a new proof of the earlier result.

scholarly journals On equational theories of semilattices with operators

1990 ◽  
Vol 42 (1) ◽  
pp. 57-70 ◽  
Author(s):  
J. Ježek ◽  
P. PudláK ◽  
J. Tůma
Keyword(s):  
Equational Theory ◽  
Algebraic Lattice ◽  
Equational Theories ◽  
Algebraic Lattices

In 1986, Lampe presented a counterexample to the conjecture that every algebraic lattice with a compact greatest element is isomorphic to the lattice of extensions of an equational theory. In this paper we investigate equational theories of semi-lattices with operators. We construct a class of lattices containing all infinitely distributive algebraic lattices with a compact greatest element and closed under the operation of taking the parallel join, such that every element of the class is isomorphic to the lattice of equational theories, extending the theory of a semilattice with operators.

scholarly journals On delta-suitable elements in algebraic lattices

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 747-754 ◽  
Author(s):  
Vanja Stepanovic ◽  
Andreja Tepavcevic
Keyword(s):  
Open Problem ◽  
New Methods ◽  
Algebraic Lattices

In this paper we introduce new methods and results that can lead to the solution of a long-standing open problem of the representation of algebraic lattices by weak congruences. Here known criteria for ?-suitable elements in algebraic lattices are generalized.

The Productivity Problem in U.S. Shipbuilding

1985 ◽  
Vol 1 (01) ◽  
pp. 7-28
Author(s):  
Bruce J. Weiers
Keyword(s):  
United States ◽  
Public Policy ◽  
Research Program ◽  
The United States ◽  
European Countries ◽  
Traditional View ◽  
Review Of The Literature ◽  
New Methods ◽  
The Difference ◽  
New View

U.S. shipbuilding productivity is significantly less than that of Japan and some European countries. The traditional view has either minimized the importance of the difference in productivity between U.S. and the best foreign shipyards, or focused on the lack of opportunities for U.S. yards to build in long series. As a result of research since 1977—much of it conducted under the auspices of the Maritime Administration National Shipbuilding Research Program—a new view of the productivity difference has developed. Several studies have established that the productivity difference is very large. A number of studies have related this difference to new methods and systems of shipbuilding developed abroad. Based on a review of the literature, this study describes these methods and systems and examines obstacles to their adoption in the United States. Implications for public policy are discussed. Some current efforts of U.S. shipbuilders to improve productivity and Maritime Administration and Navy programs of technology promotion are referenced.

scholarly journals The quasi Scott (Lawson) topology and q-continuous (q-algebraic) complete lattices

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 193-207
Author(s):  
D.N. Georgiou ◽  
A.C. Megaritis
Keyword(s):  
Complete Lattice ◽  
Scott Topology ◽  
Complete Lattices

Let L be a complete lattice. On L we define the so called quasi Scott topology, denoted by ?qSc. This topology is always larger than or equal to the Scott topology and smaller than or equal to the strong Scott topology. Results concerning the above topology are given. Also, we introduce and investigate the notions of q-continuous and q-algebraic complete lattices. Finally, we give and examine the quasi Lawson topology on a complete lattice.

COMMUTATIVE IMPLICATIONS ON COMPLETE LATTICES

1994 ◽  
Vol 02 (03) ◽  
pp. 333-341 ◽  
Author(s):  
WANGMING WU
Keyword(s):  
Normal Form ◽  
Complete Lattice ◽  
Conjunctive Normal Form ◽  
Disjunctive Normal Form ◽  
Complete Lattices

This paper is devoted to the investigation of commutative implications on a complete lattice L. It is proved that the disjunctive normal form (DNF) of a linguistic composition * is included in the conjunctive normal form (CNF) of that *, i.e., DNF(*) ≤ CNF(*) holds, for a special family of t-norms, t-conorms and negations induced by commutative implications.

On Correspondences Between Unary Algebras, Automata, Semigroups and Congruences

2006 ◽  
Vol 13 (03) ◽  
pp. 495-506 ◽  
Author(s):  
T. Petković ◽  
M. Ćirić ◽  
S. Bogdanović
Keyword(s):  
Complete Lattice ◽  
Lattice Of Varieties ◽  
Complete Lattices ◽  
Free Semigroups

In this paper, we give correspondences between unary algebras, semigroups and congruences on free semigroups. We establish isomorphisms between the complete lattice of varieties of semigroups and the complete lattices of families of varieties of unary algebras, and families of filters of congruences on free semigroups. Similar correspondences between generalized varieties and pseudovarieties of semigroups and corresponding families of algebras and congruences are also established.

scholarly journals Compact generation in partially ordered sets

1987 ◽  
Vol 42 (1) ◽  
pp. 69-83 ◽  
Author(s):  
Marcel Erné
Keyword(s):  
Partially Ordered Sets ◽  
Decomposition Theorem ◽  
Chain Condition ◽  
Ordered Sets ◽  
Direct Sums ◽  
Complete Lattices ◽  
Algebraic Lattices ◽  
Ascending Chain Condition ◽  
Structural Aspects ◽  
Compactly Generated

AbstractSeveral “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.

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