Order-reversing Involution Research Articles

Reflexive homology

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

Measurement of Countable Compactness and Lindelöf Property in RL -Fuzzy Topological Spaces

Based on the concepts of pseudocomplement of L -subsets and the implication operator where L is a completely distributive lattice with order-reversing involution, the definition of countable RL -fuzzy compactness degree and the Lindelöf property degree of an L -subset in RL -fuzzy topology are introduced and characterized. Since L -fuzzy topology in the sense of Kubiak and Šostak is a special case of RL -fuzzy topology, the degrees of RL -fuzzy compactness and the Lindelöf property are generalizations of the corresponding degrees in L -fuzzy topology.

Degrees of L-Continuity for Mappings between L-Topological Spaces

By means of the residual implication on a frame L, a degree approach to L-continuity and L-closedness for mappings between L-cotopological spaces are defined and their properties are investigated systematically. In addition, in the situation of L-topological spaces, degrees of L-continuity and of L-openness for mappings are proposed and their connections are studied. Moreover, if L is a frame with an order-reversing involution ′ , where b ′ = b → ⊥ for b ∈ L , then degrees of L-continuity for mappings between L-cotopological spaces and degrees of L-continuity for mappings between L-topological spaces are equivalent.

L-fuzzy rough approximation operators via three new types of L-fuzzy relations

Considering L being a frame with an order-reversing involution, three new types of L-fuzzy relations are introduced, which are called mediate, Euclidean and adjoint L-fuzzy relations, respectively. By means of these L-fuzzy relations, three types of L-fuzzy rough approximation operators are constructed and their connections with those three L-fuzzy relations are examined, respectively. An axiomatic approach is adopted to deal with L-fuzzy rough approximation operators. It is shown that each type of L-fuzzy rough approximation operators corresponding to mediate, Euclidean and adjoint L-fuzzy relations as well as their compositions can be characterized by single axioms.

CHARACTERIZATIONS OF L-CONVEX SPACES

In this paper, the concepts of $L$-concave structures, concave $L$-interior operators and concave $L$-neighborhood systems are introduced. It is shown that the category of $L$-concave spaces and the category of concave $L$-interior spaces are isomorphic, and they are both isomorphic to the category of concave $L$-neighborhood systems whenever $L$ is a completely distributive lattice. Also, it is proved that these categories are all isomorphic to the category of $L$-convex spaces whenever $L$ is a completely distributive lattice with an order-reversing involution operator.

Symmetric Heyting relation algebras with applications to hypergraphs

A relation on a hypergraph is a binary relation on the set consisting of all the nodes and the edges, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join preserving mappings on the lattice of sub-hypergraphs. This paper introduces a generalization of a relation algebra in which the Boolean algebra part is replaced by a Heyting algebra that supports an order-reversing involution. A general construction for these symmetric Heyting relation algebras is given which includes as a special case the algebra of relations on a hypergraph. A particular feature of symmetric Heyting relation algebras is that instead of an involutory converse operation they possess both a left converse and a right converse which form an adjoint pair of operations. Properties of the converses are established and used to derive a generalization of the well-known connection between converse, complement, erosion and dilation in mathematical morphology. This provides part of the foundation necessary to develop mathematical morphology on hypergraphs based on relations on hypergraphs.

A category equivalence for odd Sugihara monoids and its applications

An odd Sugihara monoid is a residuated distributive lattice-ordered commutative idempotent monoid with an order-reversing involution that fixes the monoid identity. The main theorem of this paper establishes a category equivalence between odd Sugihara monoids and relative Stone algebras. In combination with known results, it swiftly determines which varieties of odd Sugihara monoids are [strongly] amalgamable and which have the strong [or weak] epimorphism-surjectivity property. In particular, the full variety is shown to have all of these properties. The results extend, with slight modification, to the case where the algebras are bounded. Logical applications include immediate answers to some questions about projective and finite Beth definability and interpolation in the uninorm-based logic IUML, its boundless fragment and all of their extensions.

Two covering polynomials of a finite poset, with applications to root systems and ad-nilpotent ideals

In this note, we introduce two polynomials associated with a finite poset P , study their properties, and then compute these polynomials in the case, where P is one of the posets considered in [13]. Consider two statistics κ and ι on P . By definition, κ(x), x ∈ P , is the number of elements of P that are covered by x, and ι(x) is the number of elements that covers x. The generating function associated with κ (resp. ι) is the upper (resp. lower) covering polynomial of P . That is, KP(q) = ∑ x∈P q κ(x) and ǨP(q) = ∑ x∈P q . The upper covering polynomial, KP , has briefly been considered, without adjective ‘upper’, in [13, Section 5]. It immediately follows from the definition that KP(1) = ǨP(1) and K ′ P(1) = Ǩ ′ P(1). The last equality follows from the observation that both values equal the number of edges in the Hasse diagram of P . Hence KP(q)− ǨP(q) = (q − 1) DP(q) for some polynomial DP , which is said to be the deviation polynomial of P . We begin with a simple observation that KP ≡ ǨP whenever P is a distributive lattice or admits an order-reversing involution. Let ∆ be an irreducible root system, ∆ a subset of positive roots, and Π ⊂ ∆ is the set of simple roots. If ∆ is reduced, then g denotes the respective simple Lie algebra, with fixed Borel subalgebra b corresponding to ∆. We determine KP , ǨP , and DP in the following three cases: 1) P = ∆; 2)P = J(∆), the poset of dual order ideals in∆, orP = J(∆\Π), the poset of dual order ideals in∆ \Π. In the Lie algebra case, these posets are isomorphic to the poset of all and strictly positive ad-nilpotent ideals in b. These two posets are also denoted by Ad and Ad0, respectively; 3) P is the poset of Abelian ideal in J(∆), denoted Ab. Let us briefly describe our results. For ∆, it is shown that deg K∆+ = deg Ǩ∆+ 6 3 and the coefficients of q in K∆+ and Ǩ∆+ are equal. This readily implies that the deviation polynomial is quite simple; namely, D∆+(q) ≡ rk∆− 1.

Regularity and normality of ( L, M)-Fuzzy topological spaces

Abstract In this paper, regularity and normality axioms are introduced in ( L, M)-fuzzy topological spaces where L and M are completely distributive lattices with order-reversing involution. Each ( L, M)-fuzzy topological space can be regarded to be regular and normal to some degree. The relations among the four separation axioms are investigated. They are equivalent to each other in an ( L, M)-fuzzy metric space.

Fuzzy Prime Boolean Filters and Their Operations in IMTL-algebras

Some characterizations of fuzzy prime Boolean filters of IMTL-algebras are given. The lattice operations and the order-reversing involution on the set PB(M) of all fuzzy prime Boolean filters of IMTL-algebras are defined. It is showed that the set PB(M) endowed with these operations is a complete quasi-Boolean algebra (a distributive complete lattice with an order-reversing involution). It is derived that the algebra M/F, which is the set of all cosets of F, is isomorphic to the Boolean algebra {0, 1} if F is a fuzzy prime Boolean filter. By introducing an adjoint pair on PB(M), it is proved that the set PB(M) is also a residuated lattice.

COUNTABLY NEAR PS-COMPACTNESS IN L-TOPOLOGICAL SPACES

In this paper, the concept of countably near PS-compactness inL-topological spaces is introduced, where L is a completely distributive latticewith an order-reversing involution. Countably near PS-compactness is definedfor arbitrary L-subsets and some of its fundamental properties are studied.

FUZZY FILTERS ON THE RESIDUATED LATTICES

In this paper, the lattice operations and the adjoint pair on the fuzzy filters set on residuated lattices are defined, the conclusion that the fuzzy filters lattice defined as such is a distributive residuated lattice is obtained. An order-reversing involution on the fuzzy strong-prime filters sublattice is introduced. It is proved that the fuzzy strong-prime filters sublattice is a quasi-Boolean algebra.

Formula omitted]-uniform spaces versus [formula omitted]-uniform spaces

This paper studies relationships between the categories of I ( L ) -uniform spaces, L-uniform spaces and uniform spaces. We construct two adjunctions: Φ L ⊣ Ψ L between the category of I ( L ) -uniform spaces and the category of L-uniform spaces and φ L ⊣ ψ L between the category of L-uniform spaces and the category of uniform spaces (with L a complete lattice with an order-reversing involution in both cases), which with L = { 0 , 1 } and L = I = [ 0 , 1 ] , respectively, reduce to the adjunction from the category of I -uniform spaces to the category of uniform spaces investigated by Katsaras. If L is any complete lattice (not necessarily with an order-reversing involution), then so is the L-unit interval I ( L ) , and we have another adjunction, viz. φ I ( L ) ⊣ ψ I ( L ) from the category of I ( L ) -uniform spaces to the category of uniform spaces. We show that the following two factorizations hold: φ I ( L ) = Φ L ∘ φ L and ψ I ( L ) = ψ L ∘ Ψ L . When L is a meet-continuous lattice with an order-reversing involution, there is also a natural link between Φ L ⊣ Ψ L and the existing adjunction Ω L ⊣ I L from the category of I ( L ) -topological spaces to the one of L-topological spaces, via the forgetful functors. An essential tool in this paper is the theory of Galois connections. It is emphasized that no kind of distributivity is assumed on the lattice L in this paper.

The category of pointwise S-proximity spaces

First it is proved that the category of pointwise quasi-uniform spaces and pointwise quasi-uniformly continuous morphisms is topological. Then the concept of pointwise S-quasi-proximity on lattice-valued fuzzy set theory is introduced in such a way as to be compatible with pointwise quasi-uniformity and pointwise p.q. metric. Each co-topology can be induced by a pointwise S-quasi-proximity. When a valued lattice is a completely distributive lattices equipped with an order-reversing involution, the concept of pointwise S-proximity can be defined by means of pointwise S-quasi-proximity. The relation between pointwise S-(quasi-)proximities and pointwise (quasi-)uniformities is discussed. It is proved that the category of pointwise S-quasi-proximity spaces is isomorphic to the category of totally bounded pointwise quasi-uniform spaces. The category of pointwise S-proximity spaces is isomorphic to a subcategory of totally bounded pointwise uniform spaces. Hence they are topological.

Urysohn closedness on completely distributive lattices

The purpose of this paper is to investigate the Urysohn closedness of co-topology on lattices which are not required to have an order-reversing involution. The concept of Urysohn closedness on lattices is introduced and characterizations of Urysohn closedness are given in terms of ideals and nets. These characterizations are obtained mainly through the introduction of a kind of convergence for ideals and nets that we call L-Urysohn convergence.

Higher separation axioms in L-topologically generated [formula omitted]-topological spaces

The main result of this paper is a theorem about inserting a pair of semicontinuous L-real-valued functions which extends the insertion theorem of Kubiak [Comment. Math. Univ. Carolinae 34 (1993) 357–362] from L={0,1} to an arbitrary meet-continuous lattice L (endowed with an order-reversing involution). With this result it is shown that the normality-type separation axioms in TOP (L) are preserved by the functor which takes an L-topological space X to the I(L) -topological space Ω L(X) obtained by providing the set X with the I(L) -topology consisting of all lower semicontinuous functions from X to I(L) . The same is proved for the case of the regularity axiom.

The α-levels of a fuzzy uniform structure and of a fuzzy proximity

There are different notions of fuzzy uniform structures and of fuzzy proximities that have been introduced in the literature. In this paper we are interested in the fuzzy uniform structure U in the sense of Gähler et al. (1998) which is defined as some fuzzy filter and we are also interested in the fuzzy proximity N in the sense of Gähler et al. (1998), called the fuzzy proximity of the internal type that is defined by means of another notion of symmetry not depending on an order-reversing involution. Here, we introduce the α-level uniform structure U α and the α-level proximity N α U and N , respectively. Weshow that there is one-to-one correspondence between a fuzzyuniform structure U and the family ( U α ) α∈ L 0 of uniform structuresthat fulfills certain conditions, is given by: U α= U α and U(U)=⋁ A∈U α,χA⩽u α . We also show that the topologies T u α and T N α associated with U α and N α coincides with the α-level topologies of the fuzzy topologies τ U and τ N associated to U and N ,respectively, that is, T U α =(τ U ) α and T N α =(τ N ) α . Moreover, we assign for each fuzzyuniform structure U an associated fuzzy proximity of theinternal type N U and hence we get the relationbetween the α-levels of U and of N U which is given by: N U α =( N U ) α .

The theory of global fuzzy neighborhood structures (II). Fuzzy topogenous orders

This paper deals with fuzzy topogenous orders, in particular, with fuzzy topogenous structures and with the more special fuzzy proximities. These structures have been investigated ay Katsaras and Petalas (1983, 1984). The notion of fuzzy proximity was introduced by Katsaeas (1980). Fuzzy topogenous structures and fuzzy proximities are represented in this paper as global fuzzy neighborhood structures. Fuzzy topogenous orders, in general, are characterized by the notion of global fuzzy neighborhood prestructure, which is a weakening of that one of global fuzzy neighborhood structure. In this paper, moreover, a modification of the notion of fuzzy proximity is considered, called fuzzy proximity of the internal type. Whereas the notion of fuzzy proximity proposed by Katsaras depends on a fixed order-reversing involution of the related lattice L, the notion of fuzzy proximity of the internal type is independent on such an involution. The investigations intthis paper demonstrate that there are important global fuzzy neighborhood structures and prestructures different from fuzzy topologies and fuzzy pretopologies, respectively.

Lattice valued completely topological generated spaces

For the value set I = [0, 1], the concepts of completely lower semicontinuous mappings and the completely induced space (which will be called completely topological generated space in present paper) were first introduced by Bhaumik and Mukherjee. When the value set is fuzzy lattice (i.e., completely distributive lattice with order-reversing involution ′), we discuss the properties of these concepts, and give many characterizations of them. Using the underlying space ( X, [ δ]) of an L-fts ( L X , δ) we answer the question when an L-fts can be completely topological generated by a crisp topological space.

L-fuzzy topological groups

In this paper, we propose the concept of L-fuzzy topological groups, and discuss their basic properties. Here L is a completely distributive lattice with order-reversing involution.