Duality for Semilattice Representations A.B
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ON F-DERIVATIONS from SEMILATTICES to LATTICES
Commun. Korean Math. Soc. 29 (2014), No. 1, pp. 27–36 http://dx.doi.org/10.4134/CKMS.2014.29.1.027 ON f-DERIVATIONS FROM SEMILATTICES TO LATTICES Yong Ho Yon and Kyung Ho Kim Abstract. In this paper, we introduce the notion of f-derivations from a semilattice S to a lattice L, as a generalization of derivation and f- derivation of lattices. Also, we define the simple f-derivation from S to L, and research the properties of them and the conditions for a lattice L to be distributive. Finally, we prove that a distributive lattice L is isomorphic to the class SDf (S,L) of all simple f-derivations on S to L for every ∧-homomorphism f : S → L such that f(x0) ∨ f(y0) = 1 for ∼ some x0,y0 ∈ S, in particular, L = SDf (S,L) for every ∧-homomorphism f : S → L such that f(x0) = 1 for some x0 ∈ S. 1. Introduction In some of the literature, authors investigated the relationship between the notion of modularity or distributivity and the special operators on lattices such as derivations, multipliers and linear maps. The notion and some properties of derivations on lattices were introduced in [10, 11]. Sz´asz ([10, 11]) characterized the distributive lattices by multipliers and derivations: a lattice is distributive if and only if the set of all meet- multipliers and of all derivations coincide. In [5] it was shown that every derivation on a lattice is a multiplier and every multiplier is a dual closure. Pataki and Sz´az ([9]) gave a connection between non-expansive multipliers and quasi-interior operators. -
Compact Topologically Torsion Elements of Topological Abelian Groups Peter Loth Sacred Heart University, [email protected]
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Sacred Heart University: DigitalCommons@SHU Sacred Heart University DigitalCommons@SHU Mathematics Faculty Publications Mathematics 2005 Compact Topologically Torsion Elements of Topological Abelian Groups Peter Loth Sacred Heart University, [email protected] Follow this and additional works at: http://digitalcommons.sacredheart.edu/math_fac Part of the Algebra Commons, and the Set Theory Commons Recommended Citation Loth, P. (2005). Compact topologically torsion elements of topological abelian groups. Rendiconti del Seminario Matematico della Università di Padova, 113, 117-123. This Peer-Reviewed Article is brought to you for free and open access by the Mathematics at DigitalCommons@SHU. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of DigitalCommons@SHU. For more information, please contact [email protected], [email protected]. REND. SEM. MAT. UNIV. PADOVA, Vol. 113 (2005) Compact Topologically Torsion Elements of Topological Abelian Groups. PETER LOTH (*) ABSTRACT - In this note, we prove that in a Hausdorff topological abelian group, the closed subgroup generated by all compact elements is equal to the closed sub group generated by all compact elements which are topologically p-torsion for some prime p. In particular, this yields a new, short solution to a question raised by Armacost [A]. Using Pontrjagin duality, we obtain new descriptions of the identity component of a locally compact abelian group. 1. Introduction. All considered groups in this paper are Hausdorff topological abelian groups and will be written additively. Let us establish notation and ter minology. The set of all natural numbers is denoted by N, and P is the set of all primes. -
Z-Continuous Posets
DISCRETE MATHEMATICS ELSEVIERI Discrete Mathematics 152 (1996) 33-45 Z-continuous posets Andrei Baranga Department ofMathematics, University ofBucharest, Str. Academiei 14, 70109, Bucharest, Romania Received 19 July 1994 Abstract The concept of 'subset system on the category Po of posets (Z-sets) was defined by Wright, Wagner and Thatcher in 1978. The term Z-set becomes meaningful if we replace Z by 'directed', 'chain', 'finite'. At the end of the paper (Wright et al. 1978), the authors suggested an attempt to study the generalized counterpart of the term 'continuous poset (lattice)' obtained by replacing directed sets by Z-sets, Z being an arbitary subset system on Po. We present here some results concerning this investigation. These results are generalized counterparts of some purely order theoretical facts about continuous posets. Keywords: Poset; Subset system; Z-set; Continuous poset; Continuous lattice; Algebraic poset; Algebraic lattice; Moore family; Closure operator; Ideal; Z-inductive poset Introduction The concept of 'subset system on the category Po of posets' (Z-sets) was defined by Wright et al. in [12]. The term Z-set becomes meaningful if we replace Z by 'directed', 'chain', 'finite', etc. On the other hand, the concept of continuous lattice was defined by Scott in [9] in a topological manner as a mathematical tool in computation theory (see [10, 11]) although it has appeared in other fields of mathematics as well, such as general topology, analysis, algebra, category theory, logic and topological algebra (see [4]), This concept was defined later in purely order theoretical terms (see [8]) and now this definition has became the one used in almost all references. -
Completeness and Compact Generation in Partially Ordered Sets
Available online at www.isr-publications.com/jmcs J. Math. Computer Sci. 16 (2016), 69{76 Research Article Completeness and compact generation in partially ordered sets A. Vaezia, V. Kharatb aDepartment of Mathematics, University of Mazandaran, P. O. Box 95447, Babolsar, Iran. bDepartment of Mathematics, University of Pune, Pune 411007, India. Abstract In this paper we introduce a notion of density in posets in a more general fashion. We also introduce completeness in posets and study compact generation in posets based on such completeness and density. c 2016 All rights reserved. Keywords: U-density, U-complete poset, U-compactly generated poset, U-regular interval. 1. Introduction We begin with the necessary definitions and terminologies in a poset P . An element x of a poset P is an upper bound of A ⊆ P if a ≤ x for all a 2 A. A lower bound is defined dually. The set of all upper bounds of A is denoted by Au (read as, A upper cone), where Au = fx 2 P : x ≤ a for every a 2 Ag and dually, we have the concept of a lower cone Al of A. If P contains a finite number of elements, it is called a finite poset. A subset A of a poset P is called a chain if all the elements of A are comparable. A poset P is said to be of length n, where n is a natural number, if there is a chain in P of length n and all chains in P are of length n. A poset P is of finite length if it is of length n for some natural number n. -
ORDER-TOPOLOGICAL LATTICES ' by MARCEL ERNE
ORDER-TOPOLOGICAL LATTICES ' by MARCEL ERNE (Received 31 October, 1978) 1. Introduction and basic concepts. The observation that convergence of real sequ- ences may be defined in terms of limits inferior and limits superior as by means of neighbourhoods in the Euclidean topology leads to the question: for which lattices does order convergence coincide with convergence in the order topology? This problem has been attacked by D. C. Kent [10], A. Gingras [7] and others. We hope to present a satisfactory solution in this paper. Although there are known several characterizations of lattices, with topological order convergence (cf. Propositions 1, 2), an evaluation of these criteria already requires some knowledge of the order topology of the given lattice. In the i present paper, we establish a purely lattice-theoretical description of those lattices for which order convergence is not only topological, but moreover, the lattice operations are continuous. Henceforth, such lattices will be referred to as order-topological lattices. All convergence statements will be formulated in terms of filters rather than nets. For an introduction to convergence functions, the reader may consult D. C. Kents's paper [9]. Let L be any lattice, partially ordered by an order relation <. For a subset Y of L, let Yl and YT denote the set of all lower and upper bounds for Y, respectively. If x is the join (i.e. the least upper bound) of Y then we indicate this by the symbol x = V Y. Similarly, we write x = A, Y if x is the meet (i.e. the greatest lower bound) of Y. -
Semilattice Sums of Algebras and Mal'tsev Products of Varieties
Mathematics Publications Mathematics 5-20-2020 Semilattice sums of algebras and Mal’tsev products of varieties Clifford Bergman Iowa State University, [email protected] T. Penza Warsaw University of Technology A. B. Romanowska Warsaw University of Technology Follow this and additional works at: https://lib.dr.iastate.edu/math_pubs Part of the Algebra Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ math_pubs/215. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Mathematics at Iowa State University Digital Repository. It has been accepted for inclusion in Mathematics Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Semilattice sums of algebras and Mal’tsev products of varieties Abstract The Mal’tsev product of two varieties of similar algebras is always a quasivariety. We consider when this quasivariety is a variety. The main result shows that if V is a strongly irregular variety with no nullary operations, and S is a variety, of the same type as V, equivalent to the variety of semilattices, then the Mal’tsev product V ◦ S is a variety. It consists precisely of semilattice sums of algebras in V. We derive an equational basis for the product from an equational basis for V. However, if V is a regular variety, then the Mal’tsev product may not be a variety. We discuss examples of various applications of the main result, and examine some detailed representations of algebras in V ◦ S. -
Steps in the Representation of Concept Lattices and Median Graphs Alain Gély, Miguel Couceiro, Laurent Miclet, Amedeo Napoli
Steps in the Representation of Concept Lattices and Median Graphs Alain Gély, Miguel Couceiro, Laurent Miclet, Amedeo Napoli To cite this version: Alain Gély, Miguel Couceiro, Laurent Miclet, Amedeo Napoli. Steps in the Representation of Concept Lattices and Median Graphs. CLA 2020 - 15th International Conference on Concept Lattices and Their Applications, Sadok Ben Yahia; Francisco José Valverde Albacete; Martin Trnecka, Jun 2020, Tallinn, Estonia. pp.1-11. hal-02912312 HAL Id: hal-02912312 https://hal.inria.fr/hal-02912312 Submitted on 5 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Steps in the Representation of Concept Lattices and Median Graphs Alain Gély1, Miguel Couceiro2, Laurent Miclet3, and Amedeo Napoli2 1 Université de Lorraine, CNRS, LORIA, F-57000 Metz, France 2 Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France 3 Univ Rennes, CNRS, IRISA, Rue de Kérampont, 22300 Lannion, France {alain.gely,miguel.couceiro,amedeo.napoli}@loria.fr Abstract. Median semilattices have been shown to be useful for deal- ing with phylogenetic classication problems since they subsume me- dian graphs, distributive lattices as well as other tree based classica- tion structures. Median semilattices can be thought of as distributive _-semilattices that satisfy the following property (TRI): for every triple x; y; z, if x ^ y, y ^ z and x ^ z exist, then x ^ y ^ z also exists. -
Measure Algebras of Semilattices with Finite Breadth
Pacific Journal of Mathematics MEASURE ALGEBRAS OF SEMILATTICES WITH FINITE BREADTH JIMMIE DON LAWSON,JOHN ROBIE LIUKKONEN AND MICHAEL WILLIAM MISLOVE Vol. 69, No. 1 May 1977 PACIFIC JOURNAL OF MATHEMATICS Vol. 69, No. 1, 1977 MEASURE ALGEBRAS OF SEMILATTIGES WITH FINITE BREADTH J. D. LAWSON, J. R. LIUKKONEN AND M. W. MlSLOVE The main result of this paper is that if S is a locally compact semilattice of finite breadth, then every complex homomorphism of the measure algebra M(S) is given by integration over a Borel filter (subsemilattice whose comple- ment is an ideal), and that consequently M(S) is a P-algebra in the sense of S. E. Newman. More generally it is shown that if S is a locally compact Lawson semilattice which has the property that every bounded regular Borel measure is concentrated on a Borel set which is the countable union of compact finite breadth subsemilattices, then M(S) is a P-algebra. Furthermore, complete descriptions of the maxi- mal ideal space of M(S) and the structure semigroup of M(S) are given in terms of S, and the idempotent and invertible measures in M(S) are identified. In earlier work Baartz and Newman have shown that if S is the finite product of totally ordered locally compact semilattices, then every complex homomorphism is given by integration over a Borel subsemilattice whose complement is an ideal [1, Th. 3.15], and consequently, the structure semigroup of M(S) in the sense of Taylor [10] is itself a semilattice [9, Th. 3]. In both papers it is shown that such results do not hold for the infinite dimensional cube S — Iω, and Newman conjectures that what is needed for these results to hold is a "finite dimensionality" condition. -
The Strength of the Gr\" Atzer-Schmidt Theorem
The Strength of the Gr¨atzer-Schmidt theorem Katie Brodhead Mushfeq Khan Bjørn Kjos-Hanssen William A. Lampe Paul Kim Long V. Nguyen Richard A. Shore January 8, 2019 Abstract The Gr¨atzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that 1 1. the set of indices of computable lattices that are complete is Π1-complete; 1 2. the set of indices of computable lattices that are algebraic is Π1-complete; 1 1 3. the set of compact elements of a computable lattice is Π1 and can be Π1-complete; and 0 4. the set of compact elements of a distributive computable lattice is Π3, and there is an algebraic 0 distributive computable lattice such that the set of its compact elements is Π3-complete. Keywords: lattice theory, computability theory. Contents 1 Introduction 2 2 Computability-theoretic analysis of lattice theoretic concepts 2 1 2.1 Index set of complete lattices is Π1-complete . .2 1 2.2 Compact elements of a lattice can be Π1-complete . .3 1 2.3 Index set of algebraic lattices is Π1-complete . .5 3 Lattices of equivalence relations 6 3.1 Congruence lattices . .7 3.2 Principal congruences can be Turing complete . .8 arXiv:1408.2167v2 [math.LO] 4 Jan 2019 4 Reverse mathematics 8 5 Compact elements in algebraic lattices of restricted kinds 9 5.1 Distributive lattices . .9 5.2 Modular lattices . 16 1 1 Introduction The Gr¨atzer-Schmidt theorem [3], also known as the congruence lattice representation theorem, states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. -
Morphisms on Closure Spaces and Moore Spaces B
International Journal of Pure and Applied Mathematics Volume 91 No. 2 2014, 197-207 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v91i2.6 ijpam.eu MORPHISMS ON CLOSURE SPACES AND MOORE SPACES B. Venkateswarlu1 §, R. Vasu Babu2, Getnet Alemu3 1Department of Mathematics GIT, GITAM University Visakhapatnam, 530 045, A.P., INDIA 2Department of Mathematics Shri Vishnu Engineering college for Women Bhimavaram, 534201, W.G. Dist., A.P., INDIA 3Department of Mathematics Gondar University Gondar, ETHIOPIA Abstract: We discuss an equivalence of the concepts of complete lattices, Moore classes, and a one-to-one correspondence between Moore classes of sub- sets of a set X and closure operators on X. Also, we establish a correspon- dence between closure operators on sets and complete lattices. We describe morphisms among partially ordered sets, lattices, Moore classes, closure opera- tors and complete lattices and discuss certain inter-relationships between these objects. AMS Subject Classification: 06A15, 06F30, 54H12 Key Words: closure operator, Moore class, compact element, complete lat- tice, algebraic Lattice, frame, morphisms 1. Introduction and Preliminaries A partially ordered set (poset) is a pair (X, ≤), where X is a non empty set and c 2014 Academic Publications, Ltd. Received: September 26, 2013 url: www.acadpubl.eu §Correspondence author 198 B. Venkateswarlu, R.V. Babu, G. Alemu ≤ is a partial order (a reflexive, transitive and antisymmetric binary relation) on X. For any subset A of X and x ∈ X, x is called a lower bound (upper bound) of A if x ≤ a (a ≤ x respectively) for all a ∈ A. -
Congruence Lattices of Semilattices
PACIFIC JOURNAL OF MATHEMATICS Vol. 49, No. 1, 1973 CONGRUENCE LATTICES OF SEMILATTICES RALPH FREESE AND J. B. NATION The main result of this paper is that the class of con- gruence lattices of semilattices satisfies no nontrivial lattice identities. It is also shown that the class of subalgebra lattices of semilattices satisfies no nontrivial lattice identities. As a consequence it is shown that if 5^* is a semigroup variety all of whose congruence lattices satisfy some fixed nontrivial lattice identity, then all the members of 5^" are groups with exponent dividing a fixed finite number. Given a variety (equational class) J^ of algebras, among the inter- esting questions we can ask about the members of SίΓ is the following: does there exist a lattice identity δ such that for each algebra A e S?~, the congruence lattice Θ(A) satisfies S? In the case that 5ίΓ has dis- tributive congruences, many strong conclusions can be drawn about the algebras of J%Γ [1, 2, 7]. In the case that 3ίΓ has permutable con- gruences or modular congruences, there is reason to hope that some similar results may be obtainable [4, 8]. A standard method of proving that a class of lattices satisfies no nontrivial lattice identities is to show that all partition lattices (lattices of equivalence relations) are contained as sublattices. The lattices of congruences of semilattices, however, are known to be pseudo-complemented [9]. It follows that the partition lattice on a three-element set (the five-element two-dimensional lattice) is not isomorphic to a sublattice of the congruence lattice of a semi- lattice, and in fact is not a homomorphic image of a sublattice of the congruence lattice of a finite semilattice. -
On Lattices and Their Ideal Lattices, and Posets and Their Ideal Posets
This is the final preprint version of a paper which appeared at Tbilisi Math. J. 1 (2008) 89-103. Published version accessible to subscribers at http://www.tcms.org.ge/Journals/TMJ/Volume1/Xpapers/tmj1_6.pdf On lattices and their ideal lattices, and posets and their ideal posets George M. Bergman 1 ∗ 1 Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA E-mail: [email protected] Abstract For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P; including the empty set, and let id(P ) = Id(P )−f?g: This note obtains various results to the effect that Id(P ) is always, and id(P ) often, \essentially larger" than P: In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone map) from Id(P ) into P; and, going the other way, that an upper semilattice P admits no semilattice homomorphism from any subsemilattice of itself onto Id(P ): The slightly smaller object id(P ) is known to be isomorphic to P if and only if P has ascending chain condition. This result is strength- ened to say that the only posets P0 such that for every natural num- n ber n there exists a poset Pn with id (Pn) =∼ P0 are those having ascending chain condition. On the other hand, a wide class of cases is noted where id(P ) is embeddable in P: Counterexamples are given to many variants of the statements proved.