Distributive Lattices

In the present paper, a semimodule M over a semiring R is called absolutely pure if it is pure in every semimodule containing it as a subsemimodule. Some well-known properties of absolutely pure modules are extended to semimodules. We introduce and study two particular subclasses of absolutely pure semimodules, namely strongly absolutely pure (SAP) and …nitely injective (f -injective) semimodules. When the semiring R is additively idempotent ,the SAP R semimodules are exactly the f -injective semimodules.A characterization of Fieldhouse regular semimodules is obtained.

The concept of Ideals and filters on implication algebra is introduced. Using the idea of upper sets we investigate basic ideas of ideals and filters in an implication algebra. Finally, Self distributive implication algebra, basic concepts on implication algebras are explained and important theorems have been proved.

A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of up-sets of the poset. A ∆1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. We show that ∆1-completions are in one-to-one correspondence with certain triples consisting of a closure system of down-sets of the poset, a closure system of up-sets of the poset, and a binary relation between these two systems. Certain ∆1-completions, which we call compact, may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact ∆1-completions of a poset include its MacNeille completion and all its join-and all its meetcompletions. These completions also include the canonical extension of the given poset, a completion that encodes the topological dual of the poset when it has one. Finally, we use our parametric description of ∆1-completions to compare the canonical extension to other compact ∆1-completions identifying its relative merits.

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Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic over L κ + ,κ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel in [Nad78] for intuitionistic logic over L ω1,ω. When κ is a regular cardinal such that κ <κ = κ, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over L κ + ,κ,κ , the language with disjunctions of at most κ formulas, conjunctions of less than κ formulas and quantification on less than κ many variables. In particular, this applies to any regular cardinal under the Generalized Continuum Hypothesis.

In this work, injective semimodule has been generalized to almost-injective semimodule. The aim of this research is to study the basic properties of the concept almost-injective semimodules. The semimodule ℳ is called almost-injective semimodule if, for each subsemimodule A of and each homomorphism : A→ ℳ, either there exists a homomorphism such that =. Or there exists a homomorphism : ℳ →Y such that = , where Y is nonzero direct summand of , and is the projection map. A semimodule ℳ is almost injective semimodule if it is almost injective relative to all semimodules. Every injective semimodule is almost injective semimodule, if ℳ is almost-injective semimodule and is simple, then ℳ is-injective. In addition, some related concepts it have been studied and investigated as well.

In this paper, the terms, simple ternary Γ-semiring, semi-simple, semisimple ternary Γ-semiring are introduced. It is proved that (1) If T is a left simple ternary Γ-semiringor a lateral simple ternary Γ-semiring or a right simple ternary Γ-semiring then T is a simple ternary Γ-semiring. (2) A ternary Γ-semiring T is simple ternary Γsemiring if and only if TΓTΓaΓTΓT = T for all a T. (3) A ternary Γ-semiring T is regular then every principal ternary Γ-ideal of T is generated by an idempotent. (4) An element a of a ternary Γ-semiring T is said to be semi simple if a

The basis of this paper is to study the concept of almost projective semimodules as a generalization of projective semimodules. Some of its characteristics have been discussed, as well as some results have been generalized from projective semimodules.

We establish a novel connection between two research areas in nonclassical logics which have been developed independently of each other so far: on the one hand, input/output logic, introduced within a research program developing logical formalizations of normative reasoning in philosophical logic and AI; on the other hand, subordination algebras, investigated in the context of a research program integrating topological, algebraic, and duality-theoretic techniques in the study of the semantics of modal logic. Specifically, we propose that the basic framework of input/output logic, as well as its extensions, can be given formal semantics on (slight generalizations of) subordination algebras. The existence of this interpretation brings benefits to both research areas: on the one hand, this connection allows for a novel conceptual understanding of subordination algebras as mathematical models of the properties and behaviour of norms; on the other hand, thanks to the well developed connection between subordination algebras and modal logic, the output operators in input/output logic can be given a new formal representation as modal operators, whose properties can be explicitly axiomatised in a suitable language, and be systematically studied by means of mathematically established and powerful tools.

A join-completion of a poset is a completion for which each element is obtainable as a supremum, or join, of elements from the original poset. It is well known that the join-completions of a poset are in one-to-one correspondence with the closure systems on the lattice of upsets of the poset. A ∆1-completion of a poset is a completion for which, simultaneously, each element is obtainable as a join of meets of elements of the original poset and as a meet of joins of elements from the original poset. We show that ∆1-completions are in one-to-one correspondence with certain triples consisting of a closure system of down-sets of the poset, a closure system of upsets of the poset, and a binary relation between these two systems. Certain ∆1-completions, which we call compact, may be described just by a collection of filters and a collection of ideals, taken as parameters. The compact ∆1-completions of a poset include its MacNeille completion and all its join-and all its meetcompletions. These completions also include the canonical extension of the given poset, a completion that encodes the topological dual of the poset when it has one. Finally, we use our parametric description of ∆1-completions to compare the canonical extension to other compact ∆1-completions identifying its relative merits.

We discuss an ongoing line of research in the relational (non topological) semantics of non-distributive logics. The developments we consider are technically rooted in dual characterization results and insights from unified correspondence theory. However, they also have broader, conceptual ramifications for the intuitive meaning of non-distributive logics, which we explore.

We present a duality for the intuitionistic modal logic IK introduced by Fischer Servi in [8, 9]. Unlike other dualities for IK reported in the literature (see for example [13]), the dual structures of the duality presented here are ordered topological spaces endowed with just one extra relation, which is used to define the set-theoretic representation of both 2 and 3. Also, this duality naturally extends the definitions and techniques used by Fischer Servi in the proof of completeness for IK via canonical model construction [10]. We also give a parallel presentation of dualities for the intuitionistic modal logics IntK2 and IntK3. Finally, we turn to the intuitionistic modal logic MIPC, which is an axiomatic extension of IK, and we give a very natural characterization of the dual spaces for MIPC introduced in [2] as a subcategory of the category of the dual spaces for

We describe and classify countable Boolean rings (which may or may not have a multiplicative identity) with finitely many distinguished ideals whose elementary theory is countably categorical. This extends the description by Macintyre and Rosenstein and subsequent authors of countably categorical Boolean algebras with finitely many distinguished ideals. Following Pierce, we take a topological approach using the language of PO systems (partially ordered sets with a distinguished subset) and topological Boolean algebras. We provide two different classifications via invariants that uniquely determine the isomorphism type: one using finite PO systems and the other using finite posets. We discuss how our findings link with previous results, but the paper is otherwise self-contained.

In the first section of this paper, we introduce the notions of fractional and invertible ideals of semirings. We also define Prüfer semirings and prove that a semiring S is Prüfer iff the semiring S is a multiplicatively cancellative semiring and every 2generated ideal of S is invertible. In this section, we also show that if S is an MC semiring, then S is a Prüfer semiring iff (A + B)(A ∩ B) = AB for all ideals A, B of S. In the second section, we give a semiring version for the Gilmer-Tasng Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last we end this paper by giving a plenty of examples of proper Gaussian and Prüfer semirings.

There exist two known types of ultrafilter extensions of first-order models, both in a certain sense canonical. One of them [16] comes from modal logic and universal algebra, and in fact goes back to [20]. Another one [27, 28] comes from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups [17] as its main precursor. By a classical fact of general topology, the space of ultrafilters over a discrete space is its largest compactification. The main result of [27, 28], which confirms a canonicity of this extension, generalizes this fact to discrete spaces endowed with an arbitrary first-order structure. An analogous result for the former type of ultrafilter extensions was obtained in [31]. Results of such kind are referred to as extension theorems. After a brief introduction, we offer a uniform approach to both types of extensions based on the idea to extend the extension procedure itself. We propose a generalization of the standard concept of first-order interpretations in which functional and relational symbols are interpreted rather by ultrafilters over sets of functions and relations than by functions and relations themselves, and define ultrafilter models with an appropriate semantics for them. We provide two specific operations which turn ultrafilter models into ordinary models, establish necessary and sufficient conditions under which the latter are the two canonical ultrafilter extensions of some ordinary models, and obtain a topological characterization of ultrafilter models. We generalize a restricted version of the extension theorem to ultrafilter models. To formulate the full version, we propose a wider concept of ultrafilter models with their semantics based on limits of ultrafilters, and show that the former concept can be identified, in a certain way, with a particular case of the latter; moreover, the new concept absorbs the ordinary concept of models. We provide two more specific operations which turn ultrafilter models in the narrow sense into ones in the wide sense, and establish necessary and sufficient conditions under which ultrafilter models in the wide sense are the images of ones in the narrow sense under these operations, and also are two canonical ultrafilter extensions of some ordinary models. Finally, we establish three full versions of the extension theorem for ultrafilter models in the wide sense. The results of the first three sections of this paper were partially announced in [25].

A subresiduated lattice is a pair (A, D), where A is a bounded distributive lattice, D is a bounded sublattice of A and for every a, b ∈ A there is c ∈ D such that for all d ∈ D, d ∧ a ≤ b if and only if d ≤ c. This c is denoted by a → b. This pair can be regarded as an algebra ⟨A, ∧, ∨, →, 0, 1⟩ of type (2, 2, 2, 0, 0) where D = {a ∈ A : 1 → a = a}. The class of subresiduated lattices is a variety which properly contains to the variety of Heyting algebras. In this paper we present dual equivalences for the algebraic category of subresiduated lattices. More precisely, we develop a spectral style duality and a bitopological style duality for this algebraic category. Finally we study the connections of these results with a known Priestley style duality for the algebraic category of subresiduated lattices.

In this paper, among other results, there are described (complete) simple-simultaneously ideal-and congruence-simple-endomorphism semirings of (complete) idempotent commutative monoids; it is shown that the concepts of simpleness, congruence-simpleness and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; there are described congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of 'Morita equivalence' and 'simpleness' in the semiring setting, we have obtained the following results: The ideal-simpleness, congruencesimpleness and simpleness of semirings are Morita invariant properties; A complete description of simple semirings containing the infinite element; The representation theorem-"Double Centralizer Property"-for simple semirings; A complete description of simple semirings containing a projective minimal onesided ideal; A characterization of ideal-simple semirings having either infinite elements or a projective minimal one-sided ideal; A confirmation of Conjecture of [18] and solving Problem 3.9 of [17] in the classes of simple semirings containing either infinite elements or projective minimal left (right) ideals, showing, respectively, that semirings of those classes are not perfect and the concepts of 'mono-flatness' and 'flatness' for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, latticeordered semirings.

We establish a novel connection between two research areas in nonclassical logics which have been developed independently of each other so far: on the one hand, input/output logic, introduced within a research program developing logical formalizations of normative reasoning in philosophical logic and AI; on the other hand, subordination algebras, investigated in the context of a research program integrating topological, algebraic, and duality-theoretic techniques in the study of the semantics of modal logic. Specifically, we propose that the basic framework of input/output logic, as well as its extensions, can be given formal semantics on (slight generalizations of) subordination algebras. The existence of this interpretation brings benefits to both research areas: on the one hand, this connection allows for a novel conceptual understanding of subordination algebras as mathematical models of the properties and behaviour of norms; on the other hand, thanks to the well developed connection between subordination algebras and modal logic, the output operators in input/output logic can be given a new formal representation as modal operators, whose properties can be explicitly axiomatised in a suitable language, and be systematically studied by means of mathematically established and powerful tools.

We study several kinds of distributivity for concept lattices of contexts. In particular, we find necessary and sufficient conditions for a concept lattice to be (1) distributive, (2) a frame (locale, complete Heyting algebra), (3) isomorphic to a topology, (4) completely distributive, (5) superalgebraic (i.e., algebraic and completely distributive). In cases (2), (4) and (5), our criteria are first order statements on objects and attributes of the given context. Several applications are obtained by considering the completion by cuts and the completion by lower ends of a quasiordered set as special types of concept lattices. Various degrees of distributivity for concept lattices are expressed by certain "separation axioms" for the underlying contexts. Passing to complementary contexts makes some statements and proofs more elegant. For example, it leads to a one-to-one correspondence between completely distributive lattices and so-called Cantor lattices, and it establishes an equivalence between partially ordered sets and doubly founded reduced contexts with distributive concept lattices.

We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consist of so-called Mammen spaces, where a Mammen space is a triple (U, S, C), where U is a non-empty set ("the universe"), S is a perfect Hausdorff topology on U , and C ⊆ P(U) together with S satisfy certain axioms. We refute a conjecture put forward by J. Hoffmann-Jørgensen, who conjectured that the existence of a "complete" Mammen space implies the Axiom of Choice, by showing that in the first Cohen model, in which ZF holds but AC fails, there is a complete Mammen space. We obtain this by proving that in the first Cohen model, every perfect topology can be extended to a maximal perfect topology. On the other hand, we also show that if all sets are Lebesgue measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe. Finally, we investigate two new cardinal invariants uM and uT associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. Further, we show uM = uT = 2 ℵ 0 follows from Martin's Axiom, and, contrastingly, we show that ℵ1 = uM = uT < 2 ℵ 0 = ℵ2 in the Baumgartner-Laver model.

Rough set theory has a vital role in the mathematical field of knowledge representation problems. Hence, a Rough algebraic structure is defined by Pawlak. Mathematics and Computer Science have many applications in the field of Lattice. The principle of the ordered set has been analyzed in logic programming for crypto-protocols. Iwinski extended an approach towards the lattice set with the rough set theory whereas an algebraic structure based on a rough lattice depends on indiscernibility relation which was established by Chakraborty. Granular means piecewise knowledge, grouping with similar elements. The universe set was partitioned by an indiscernibility relation to form a Granular. This structure was framed to describe the Rough set theory and to study its corresponding Rough approximation space. Analysis of the reduction of granular from the information table is based on object-oriented. An ordered pair of distributive lattices emphasize the congruence class to define its projection. This projection of distributive lattice is analyzed by a lemma defining that the largest and the smallest elements are trivial ordered sets of an index. A Rough approximation space was examined to incorporate with the upper approximation and analysis with various possibilities. The Cartesian product of the distributive lattice was investigated. A Lattice homomorphism was examined with an equivalence relation and its conditions. Hence the approximation space exists in its union and intersection in the upper approximation. The lower approximation in different subsets of the distributive lattice was studied. The generalized lower and upper approximations were established to verify some of the results and their properties.

In this note we provide an explicit construction of F Q(n), the free Q-distributive lattice over an n-element chain, different from those given by Cignoli [4] and Abad-Díaz Varela [1], and prove that F Q(n) can be endowed with a structure of a De Morgan algebra. 1-Preliminaries Quantifiers on distributive lattices were considered for the first time by Servi in [14], but it was Cignoli [3] who studied them as algebras, which he named Q-distributive lattices. In [4], Cignoli gave a construction of the free Q-distributive lattice over a set X. This construction generalizes that given by Halmos [6] for the free monadic Boolean algebra. The present paper is motivated by a result of Abad and Díaz Varela [1] on the free Q-distributive lattice, F Q(I), over a finite ordered set I. They proved that F Q(n) ∼ = 2 [2 [2×n] ] , where n is an n-element chain. We provide an easy construction for the diagram of 2 [2×n] , and we prove some properties of F Q(n). We include in this section some notation, definitions and results on distributive lattices which are used in the paper. An element p of a lattice R is said to be join-irreducible if p = 0 and p = x ∨ y implies p = x or p = y, for all x, y ∈ R. We denote the set of join-irreducible elements of R by J (R).

It is introduced a new algebra $$(A, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$ ( A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1 ) called $$L_PG$$ L P G -algebra if $$(A, \otimes , \oplus , *, 0, 1)$$ ( A , ⊗ , ⊕ , ∗ , 0 , 1 ) is $$L_P$$ L P -algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and $$(A,\rightharpoonup , 0, 1)$$ ( A , ⇀ , 0 , 1 ) is a Gödel algebra (i.e. Heyting algebra satisfying the identity $$(x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1)$$ ( x ⇀ y ) ∨ ( y ⇀ x ) = 1 ) . The lattice of congruences of an $$L_PG$$ L P G -algebra $$(A, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$ ( A , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1 ) is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra $$(A, \otimes , \oplus , *, 0, 1)$$ ( A , ⊗ , ⊕ , ∗ , 0 , 1 ) . The variety $$\mathbf {L_PG}$$ L P G of $$L_PG$$ L P G -algebras is generated by the algebras $$(C, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$ ( C , ⊗ , ⊕ , ∗ , ⇀ , 0 , 1 ) where...

We introduce a simple modal calculus for compact Hausdorff spaces. The language of our system extends that of propositional logic with a strict implication connective, which, as shown in earlier work, algebraically corresponds to the notion of a subordination on Boolean algebras. Our base system is a strict implication calculus SIC, to which we associate a variety SIA of strict implication algebras. We also study the symmetric strict implication calculus S 2 IC, which is an extension of SIC, and prove that S 2 IC is strongly sound and complete with respect to de Vries algebras. By de Vries duality, this yields completeness of S 2 IC with respect to compact Hausdorff spaces. Since some of the defining axioms of de Vries algebras are Π 2-sentences, we develop the corresponding theory of non-standard rules, which we term Π 2-rules. We study the resulting inductive elementary classes of algebras, and give a general criterion of admissibility for Π 2-rules. We also compare our approach to approaches in the literature that are related to our work. 1

The Vakarelov construction of Nelson algebras up from Heyting ones is generalized to obtain De Morgan algebras from distributive lattices. Necessary and sufficient conditions for these De Morgan algebras to be Nelson algebras are shown, and a characterization of the join-irreducible elements in the finite case is given.Fil: Monteiro, Luiz F.. Universidad Nacional del Sur; ArgentinaFil: Viglizzo, Ignacio Dario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaXIV Congreso Dr. Antonio A. R. MonteiroBahía BlancaArgentinaUniversidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática de Bahía Blanc

Rough set theory has a vital role in the mathematical field of knowledge representation problems. Hence, a Rough algebraic structure is defined by Pawlak. Mathematics and Computer Science have many applications in the field of Lattice. The principle of the ordered set has been analyzed in logic programming for crypto-protocols. Iwinski extended an approach towards the lattice set with the rough set theory whereas an algebraic structure based on a rough lattice depends on indiscernibility relation which was established by Chakraborty. Granular means piecewise knowledge, grouping with similar elements. The universe set was partitioned by an indiscernibility relation to form a Granular. This structure was framed to describe the Rough set theory and to study its corresponding Rough approximation space. Analysis of the reduction of granular from the information table is based on object-oriented. An ordered pair of distributive lattices emphasize the congruence class to define its projection. This projection of distributive lattice is analyzed by a lemma defining that the largest and the smallest elements are trivial ordered sets of an index. A Rough approximation space was examined to incorporate with the upper approximation and analysis with various possibilities. The Cartesian product of the distributive lattice was investigated. A Lattice homomorphism was examined with an equivalence relation and its conditions. Hence the approximation space exists in its union and intersection in the upper approximation. The lower approximation in different subsets of the distributive lattice was studied. The generalized lower and upper approximations were established to verify some of the results and their properties.

A commutative doubly-idempotent semiring (cdi-semiring) (S, ∨, •, 0, 1) is a semilattice (S, ∨, 0) with x ∨ 0 = x and a semilattices (S, •, 1) with identity 1 such that x0 = 0, and x(y ∨ z) = xy ∨ xz holds for all x, y, z ∈ S. Bounded distributive lattices are cdi-semirings that satisfy xy = x ∧ y, and the variety of cdi-semirings covers the variety of bounded distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdi-semirings is generated by a 3-element cdi-semiring. We show that there are seven cdi-semirings with a ∨-semilattice of height less than or equal to 2. We construct all cdi-semirings for which their multiplicative semilattice is a chain with n + 1 elements, and we show that up to isomorphism the number of such algebras is the n th Catalan number Cn = 1 n+1 (2n n). We also show that cdi-semirings with a complete atomic Boolean ∨-semilattice on the set of atoms A are determined by singleton-rooted preorder forests on the set A. From these results we obtain efficient algorithms to construct all multiplicatively linear cdisemirings of size n and all Boolean cdi-semirings of size 2 n .

In this paper, we initiate the discourse on the properties that hold in an almost semi-Heyting algebra but not in an semi-Heyting almost distributive lattice. We establish an equivalent condition for an almost semi-Heyting algebra to become a Stone almost distributive lattice. Moreover a glance about dense elements in an almost semi-Heyting algebra followed by study of some algebraic properties on them. Finally, we perceive that the kernel of homomorphism is equal to the dense element set.

The Vakarelov construction of Nelson algebras up from Heyting ones is generalized to obtain De Morgan algebras from distributive lattices. Necessary and sufficient conditions for these De Morgan algebras to be Nelson algebras are shown, and a characterization of the join-irreducible elements in the finite case is given.Fil: Monteiro, Luiz F.. Universidad Nacional del Sur; ArgentinaFil: Viglizzo, Ignacio Dario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaXIV Congreso Dr. Antonio A. R. MonteiroBahía BlancaArgentinaUniversidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática de Bahía Blanc

The Vakarelov construction of Nelson algebras up from Heyting ones is generalized to obtain De Morgan algebras from distributive lattices. Necessary and sufficient conditions for these De Morgan algebras to be Nelson algebras are shown, and a characterization of the join-irreducible elements in the finite case is given.Fil: Monteiro, Luiz F.. Universidad Nacional del Sur; ArgentinaFil: Viglizzo, Ignacio Dario. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaXIV Congreso Dr. Antonio A. R. MonteiroBahía BlancaArgentinaUniversidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática de Bahía Blanc

The generalized closure operator  induces a topology . In this paper, we study the topology  on lower bounded Alexandroff spaces. We prove that  is a submaximal Alexandroff space. We get some new results about the relation between  and . Then we prove that a subset  in a lower bounded space is closed set if and only if  is open in the dual space. v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} 14.00 Normal 0 false false false EN-US ZH-CN AR-SA /* Style Definitions */ table.MsoNormalTable {mso-style-name:&quot;جدول عادي&quot;; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:&quot;&quot;; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin-top:0cm; mso-para-margin-right:0cm; mso-para-margin-bottom:10.0pt; mso-para-margin-left:0cm; line-height:115%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:&quot;Calibri&quot...

This dissertation contains two parts: lattice theory and graph theory. In the lattice theory part, we have two main subjects. First, the class of all distributive lattices is one of the most familiar classes of lattices. We introduce "7r-versions" of five familiar equivalent conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D 0n if a A (b Vc) < (a A b) Vc for all 3-element antichains {a, b, c}. We consider a congruence relation ~ whose blocks are the maximal autonomous chains and define the orderskeleton of a lattice L to be L := L/~. We prove that the following are equivalent for a lattice L: (i) L satisfies D Qw , (ii) L satisfies any of the five 7r-versions of distributivity, (Hi) the order-skeleton L is distributive. Second, the symmetric difference notion for Boolean algebra is well-known. Matousek introduced the orthocomplemented difference lattices (ODLs), which are ortholattices associated with a symmetric difference. He proved that the class of ODLs forms a variety. We focus on the class of all ODLs that are set-representable and prove that this class is not locally finite by constructing an infinite set-representable ODL that is generated by three elements. In the graph theory part, we prove generating theorems and splitter theorems for 5-regular graphs. A generating theorem for a certain class of graphs tells us how to iii iv generate all graphs in this class from a few graphs by using some graph operations. A splitter theorem tells us how to build up any graph G from any graph HUG "contains" H. In this dissertation, we find generating theorems for 5-regular graphs and 5-regular loopless graphs for various edge-connectivities. We also find splitter theorems for 5-regular graphs for various edge-connectivities.

The article has two main objectives: characterize compact Hausdorff topological MV-algebras and Stone MV-algebras on one hand, and characterize strongly complete MV-algebras on the other hand. We obtain that compact Hausdorff topological MV-algebras are product (both topological and algebraic) of copies [0, 1] with the interval topology and finite Lukasiewicz chains with discrete topology. Going one step further we also prove that Stone MV-algebras are product (both topological and algebraic) of finite Lukasiewicz chains with discrete topology. In the second part we prove that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite and its only maximal ideals of finite ranks are principal.

Constructive meaning is given to the assertion that every finite Boolean algebra is an injective object in the category of distributive lattices. To this end, we employ Scott&#39;s notion of entailment relation, in which context we describe Sikorski&#39;s extension theorem for finite Boolean algebras and turn it into a syntactical conservation result. As a by-product, we can facilitate proofs of several related classical principles.

We study the lattice of all normal consistent extensions of the modal logic S4, NExtS4, from a structural point of view. We show that a pattern isomorphic to the lattice of intermediate logics is present as a sublattice in NExtS4 in many, in fact, in infinitely many places, and this pattern itself is isomorphic to a quotient lattice of NExtS4. We also designate three "dark spots" of NExtS4, three sublattices of it, where, although we can characterize the logics belonging to each of these sublattices, their structural picture is invisible at all.

If ( X , τ ) is a topological space, then the semi-regularization topology τ s on X of τ is the coarser topology on X generated by the family of all open domains of ( X , τ ) where a subset U is called an open domain if U = int(U). In this paper, we study the semi-regularity of some generated spaces and some properties of weaker version of normality of the semi-regularization space ( X , τ s ) of a space ( X , τ ).

Based on the theory of frames we introduce a Stone duality for bitopological spaces. The central concept is that of a d-frame, which axiomatises the two open set lattices. Exploring the resulting concept of d-sobriety we find this to be a much more inclusive concept than usual sobriety. Spatial d-frames suggest additional axioms that lead us to define reasonable d-frames; these have an alternative presentation as partial frames. We explore natural notions of regularity and compactness for bitopological spaces, and their manifestation in d-frames. This yields the machinery to locate precisely within this general landscape a number of classical Stonetype dualities, namely, those of Stone for Boolean algebras and bounded distributive lattices, those of the present authors for strong proximity lattices (with negation), and the duality of classical frames. The general duality can be given a logical reading by viewing the open sets of one topology as positive extents of formulas, and those of the other topology as negative extents. This point of view emphasises the fact that formulas may be undecidable in certain states and may be self-contradictory in others. We also obtain two natural orders on the set of formulas, one related to Scott's information order and the other being the usual logical implication. The interplay between the two can be said to be the main organising principle of this study. is exactly the complement of those that are associated with b , and so the negation operation is faithfully modelled by complement. Stone did not stop here but realised that it was just as important to represent the homomorphisms between the algebras. In order to single out the right set-theoretic maps between collections of prime filters, he employed topology. The basic opens of the Stone topology on S ¥ p are exactly the sets T ¤ ¥ b 3 4 © e 7 S ¥ p @ © b 7 9 e C which were used in the proof of the representation theorem. The topological space so obtained is called the spectrum of p (borrowing terminology from Ring Theory).

If ( X , τ ) is a topological space, then the semi-regularization topology τ s on X of τ is the coarser topology on X generated by the family of all open domains of ( X , τ ) where a subset U is called an open domain if U = int(U). In this paper, we study the semi-regularity of some generated spaces and some properties of weaker version of normality of the semi-regularization space ( X , τ s ) of a space ( X , τ ).

We observe that if R := (I, ρ, J) is an incidence structure, viewed as a matrix, then the topological closure of the set of columns is the Stone space of the Boolean algebra generated by the rows. As a consequence, we obtain that the topological closure of the collection of principal initial segments of a poset P is the Stone space of the Boolean algebra T ailalg(P) generated by the collection of principal final segments of P , the so-called tail-algebra of P. Similar results concerning Priestley spaces and distributive lattices are given. A generalization to incidence structures valued by abstract algebras is considered.

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.

In this paper, we deal with the problem of putting together modal worlds that operate in different logic systems. When evaluating a modal sentence 2φ, we argue that it is not sufficient to inspect the truth of φ in accessed worlds (possibly in different logics). Instead, ways of transferring more subtle semantic information between logical systems must be established. Thus, we will introduce modal structures that accommodate communication between logic systems by fixing a common lattice L where different logics build their semantics. The semantics of each logic being considered in the modal structure is a sublattice of L. In this system, necessity and possibility of a statement should not solely rely on the satisfaction relation in each world and the accessibility relation. The value of a formula 2φ will be defined in terms of a comparison between the values of φ in accessible worlds and the common lattice L. We will investigate natural instances where formulas φ can be said to be n...