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In order to deal with uncertainty, Molodtsov propoed soft set theory as a mathematical tool in 1999. Since that time, numerous forms of soft set operations have been defined and employed. By establishing the soft binary piecewise operations' distributions over complementary soft binary piecewise intersection and union operations, this study aims to contribute to the literature on soft set theory by giving inspiration to researchers as regards examining and obtaining some algebraic structures using these new soft set operations. [ABSTRACT FROM AUTHOR]

In this paper, we will show that based on the concept of multialgebras and relying on the power structures of these algebras, which we could also consider as relational structures, and adding the set of operations ... we get a Boolean algebra with operator. The power structure of relational structure = (A,) is an algebra with the set (A) and the set of all basic operations defined. [ABSTRACT FROM AUTHOR]

We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function $${f : \mathbb{N} \rightarrow \mathbb{N}}$$ such that the n-ary part of C is equal to the n-ary part of Pol $${{\rm Inv}^{[f(n)]} C}$$ , where $${{\rm Inv}^{[f(n)]} C}$$ denotes the set of f( n)-ary invariant relations of C. [ABSTRACT FROM AUTHOR]

In this paper, we define the sequence space BV ⌢θ (f, p, q, s) on a seminormed complex linear space, by using a Modulus function. We give various properties and some inclusion relations on this space. [ABSTRACT FROM AUTHOR]

A d-hypercube of order n is an n × ··· × nd (d times) array with nd elements from a set Q of cardinality n. We recall several connections between d-hypercubes of order n and d-ary operations of order n. We give constructions of orthogonal d-ary operations that generalize a result of Belyavskaya and Mullen. Our main result is a general construction of d-orthogonal d-ary operations from d-ary quasigroups. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the concept of soft uniform spaces and the relationships between soft uniform spaces and uniform spaces. The notions of soft uniform structure, soft uniform continious function, and operations on soft uniform space are introduced and their basic properties are investigated. [ABSTRACT FROM AUTHOR]

n this paper we extend the notions of γ-operation, pre-open msets, α-open msets, semi open msets, b-open msets and β-open msets to M-topological spaces. These types of msets are new classes of multisets. We study the rela- tions between these different types of submsets of M-topological spaces. Also, we study some of their properties and show that these types generalize the notion of open (closed) msets. [ABSTRACT FROM AUTHOR]

We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related--i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term. [ABSTRACT FROM AUTHOR]

The symmetric homology of a unital associative algebra A over a commutative ground ring k, denoted HS*(A), is defined using derived functors and the symmetric bar construction of Fiedorowicz. In this paper we show that HS*(A) admits homology operations and a Pontryagin product structure making HS*(A) an associative commutative graded algebra. This is done by finding an explicit E∞ structure on the standard chain groups that compute symmetric homology. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present an optical processor for performing arithmetic operations in parallel. The implementation of arithmetic operations makes it possible to perform various computational tasks. As case studies we solve the bounded subset sum problem in parallel and perform parallel primality testing. The processor uses two-dimensional optoelectronic planes for both performing logic operations and storing data, which eliminates the need for transferring data from electronic devices in each step of the computation. The presented processor seems easier to realize than most of the past parallel optical processors due to its simpler and more compact architecture, while staying powerful enough to carry out computations from diverse applications. [Copyright &y& Elsevier]

Let D be an integral domain with quotient field K, * be a staroperation on D, and GV *(D) be the set of finitely generated ideals J of D such that J* = D. Then the map *w defined by I*w = {x Є K | Jx ⊆ I for some J Є GV *(D)} for all nonzero fractional ideals I of D is a finite character staroperation on D. In this paper, we study several properties of *w-Noetherian domains. In particular, we prove the Hilbert basis theorem for *w-Noetherian domains. [ABSTRACT FROM AUTHOR]

Any set x is uniquely specified by the graph of the membership relation on the set obtained by adjoining x to the transitive closure of x. Thus any operation on sets can be looked at as an operation on these graphs. We look at the operations of ordinal arithmetic of sets in this light. This turns out to be simplest for a modified ordinal arithmetic based on the Zermelo ordinals, instead of the usual von Neumann ordinals. In this arithmetic, addition of sets corresponds to concatenating graphs, and multiplication corresponds to replacing each edge of a graph by a copy of another graph. Characterizations for the von Neumann case are also given. [ABSTRACT FROM AUTHOR]

LetSbe a semigroup andPbe a nonempty subset ofS. If we define a hyperoperation ○ onSbyx○y=xPyfor allx,y∈S, then (S, ○) is a semihypergroup and is denoted by (S,P) . In this paper we get conditions onSandPso that (S,P) is a regular semihypergroup. [ABSTRACT FROM PUBLISHER]

The concept of soft sets introduced by Molodtsov is a general mathematical tool for dealing with uncertainty. Just as the conventional set-theoretic operations of intersection, union, complement, and difference, some corresponding operations on soft sets have been proposed. Unfortunately, such operations cannot keep all classical set-theoretic laws true for soft sets. In this paper, we redefine the intersection, complement, and difference of soft sets and investigate the algebraic properties of these operations along with a known union operation. We find that the new operation system on soft sets inherits all basic properties of operations on classical sets, which justifies our definitions. [ABSTRACT FROM AUTHOR]

In this paper, our main tool is the use of mapping &bgr;:SO(X) ->P(X) from the semi-open set into power set P(X) of the underlying set X, having the property of mono- tonicity and A ⫅&bgr;(A) for each semi-open sets A, where &bgr;(A) denotes the value of A under &bgr;. Moreover, the operation &bgr; generalize the notions and characterizations defined and discussed using the operation &agr; defined and discussed by Kasahara[6]. [ABSTRACT FROM AUTHOR]

Abstract: We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of Bredon–Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients. [Copyright &y& Elsevier]

Let k be a global field, p an odd prime number different from char( k) and S, T disjoint, finite sets of primes of k. Let be the Galois group of the maximal p-extension of k which is unramified outside S and completely split at T. We prove the existence of a finite set of primes S0, which can be chosen disjoint from any given set ℳ of Dirichlet density zero, such that the cohomology of coincides with the étale cohomology of the associated marked arithmetic curve. In particular, . Furthermore, we can choose S0 in such a way that realizes the maximal p-extension k( p) of the local field k for all  ∈ S ∪ S0, the cup-product is surjective and the decomposition groups of the primes in S establish a free product inside . This generalizes previous work of the author where similar results were shown in the case T = ∅︀ under the restrictive assumption p ∤ #Cl( k) and ζ p ∉ k. [ABSTRACT FROM AUTHOR]

Abstract: We give a classification of e.a.b. semistar (and star) operations by defining four different (successively smaller) distinguished classes. Then, using a standard notion of equivalence of semistar (and star) operations to partition the collection of all e.a.b. semistar (or star) operations, we show that there is exactly one operation of finite type in each equivalence class and that this operation has a range of nice properties. We give examples to demonstrate that the four classes of e.a.b. semistar (or star) operations we defined can all be distinct. In particular, we solve the open problem of showing that a.b. is really a stronger condition than e.a.b. [Copyright &y& Elsevier]

The set of clones of operations on {0, 1} forms a countable lattice which was classified by Post. The cardinality of the lattice of hyperclones on {0, 1} was proved by Machida to be of the continuum. A hyperclone C1 is a hypercore of a clone C if its extension C1# is contained in C and C1 is not contained in any other hyperclone having the same property. We describe six maximal hyperclones on the two-elements set as hypercores of particular clones on three element set. The interval of hyperclones on {0, 1} between the clone of all projections and the one generated by all unary hyperoperations is also completely determined. [ABSTRACT FROM AUTHOR]

In this paper, we prove a fuzzy version of Arrow's Theorem that contains the crisp version. We show that under our definitions, Arrow's Theorem remains intact even if levels of intensities of the players and levels of membership in the set of alternatives are considered. [ABSTRACT FROM AUTHOR]

We provide an interpretation of the homology of the loop space on the $p$-completion of the classifying space of a finite group in terms of representation theory, and demonstrate how to compute it. We then give the following reformulation. If $f$ is an idempotent in $kG$ such that $f.kG$ is the projective cover of the trivial module $k$, and $e=1-f$, then we exhibit isomorphisms for $nge 2$: begin {align*} H_n(Omega BG {}^{^wedge }_p;k) &cong mathrm {Tor}_{n-1}^{e.kG.e}(kG.e,e.kG), H^n(Omega BG{}^{^wedge }_p;k) &cong mathrm {Ext}^{n-1}_{e.kG.e}(e.kG,e.kG). endalign* }}} Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations. [ABSTRACT FROM AUTHOR]

Let $(R,mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i in mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i not = c = operatorname {height} I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$. Here we prove that this is true if and only if $H^i_I(R) = 0$ for $i = n, n-1$, provided $c geq 2$ and $R/I$ has an isolated singularity, resp. if $I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing result of $H^i_I(R)$ for all $i > m, m$ a given integer, and an estimate of the dimension of $H^i_I(R).$ [ABSTRACT FROM AUTHOR]

Let (R, 픪) be a complete Noetherian local ring and let M be a finite R-module of positive Krull dimension n. It is shown that any subset T of AsshR(M) can be expressed as the set of attached primes of the top local cohomology module ${\rm H}^n_{\frak a}(M)$ for some ideal 픞 of R. Moreover, if 픞 is an ideal of R such that the set of attached primes of ${\rm H}^n_{\frak a}(M)$ is a non-empty proper subset of AsshR(M), then ${\rm H}^n_{\frak a}(M)\cong {\rm H}^n_{\frak b}(M)$ for some ideal 픟 of R with dimR(R/픟) = 1. [ABSTRACT FROM AUTHOR]

Orbispaces are the analog of orbifolds, where the category of manifolds is replaced by topological spaces. We construct the loop orbispace LX of an orbispace X in the language of stacks in topological spaces. Furthermore, to a twist given by a U(1)-banded gerbe G → X we associate a U(1)δ-principal bundle G˜δ → LX. We use sheaf theory on topological stacks in order to define the delocalized twisted cohomology by H*deloc(X,G) := 8(GL, f*L L), where fL : GL → LX is the pull-back of the gerbe G → X via the natural map LX → X, and L ϵ ShAbLX is the sheaf of sections of the ℂδ-bundle associated to G˜δ → LX. The same constructions can be applied in the case of orbifolds, and we show that the sheaf theoretic delocalized twisted cohomology is isomorphic to the twisted de Rham cohomology, where the isomorphism depends on the choice of a geometric structure on the gerbe G → X. [ABSTRACT FROM AUTHOR]

We define a notion of cofibration among ∞-categories and show that the cofibrant objects are exactly the free ones, that is, those generated by polygraphs. [ABSTRACT FROM AUTHOR]

Let X and A be sets and α : X → A a map between them. We call a map μ : X × X × X → A an approximate Mal'tsev operation with approximation α, if it satisfies μ(x, y, y) = α(x) = μ(y, y, x) for all x, y ∊ X. Note that if A = X and the approximation α is an identity map, then μ becomes an ordinary Mal'tsev operation. We prove the following two characterization theorems: a category 핏 is a Mal'tsev category if and only if in the functor category Set핏op×핏 there exists an internal approximate Mal'tsev operation hom핏× hom핏 × hom핏 → A whose approximation α satisfies a suitable condition; a regular category 핏 with finite coproducts is a Mal'tsev category, if and only if in the functor category 핏핏 there exists an internal approximate Mal'tsev cooperation A → 1핏 + 1핏 + 1핏 whose approximation α is a natural transformation with every component a regular epimorphism in 핏. Note that in both of these characterization theorems, if require further the approximation α to be an identity morphism, then the conditions there involving α become equivalent to 핏 being a naturally Mal'tsev category. [ABSTRACT FROM AUTHOR]

We study Ruan's cohomological crepant resolution conjecture [41] for orbifolds with transversal ADE singularities. In the An-case, we compute both the Chen–Ruan cohomology ring $H_{\rm CR}^{*}([Y])$ and the quantum corrected cohomology ring H*(Z)(q1,...,qn). The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between $H_{\rm CR}^{*}([Y])$ and H*(Z)(-1) in the A1-case, verifying Ruan's conjecture. In the An-case, the family H*(Z)(q1,...,qn) is not defined for q1 = ⋯ = qn = -1. This implies that the conjecture should be slightly modified. We propose a new conjecture in the An-case (Conjecture 1.9). Finally, we prove Conjecture 1.9 in the A2-case by constructing an explicit isomorphism. [ABSTRACT FROM AUTHOR]

Over a perfect field $k$ of characteristic $p > 0$, we construct a ‘Witt vector cohomology with compact supports’ for separated $k$-schemes of finite type, extending (after tensorization with ${\mathbb Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope ${<}1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre. [ABSTRACT FROM AUTHOR]

Abstract: Let C be a complex affine reduced curve, and denote by its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant that measures the complexity of the singularity of C at the point x, and we establish the following formula: where denotes the first singular homology group of C with complex coefficients. Second we consider a family of curves given by the fibres of a dominant morphism , where X is an irreducible complex affine surface. We analyze the behaviour of the function . More precisely we show that it is constant on a Zariski open set, and that it is lower semi-continuous in general. [Copyright &y& Elsevier]

A group $G$ is said to have periodic cohomology with period $q$ after $k$ steps, if the functors $H^{i}(G,-)$ and $H^{i+q}(G,-)$ are naturally equivalent for all $i>k$. Mislin and the author have conjectured that periodicity in cohomology after some steps is the algebraic characterization of those groups $G$ that admit a finite-dimensional, free $G$-CW-complex, homotopy equivalent to a sphere. This conjecture was proved by Adem and Smith under the extra hypothesis that the periodicity isomorphisms are given by the cup product with an element in $H^q (G,\mathbb{Z})$. It is expected that the periodicity isomorphisms will always be given by the cup product with an element in $H^q (G,\mathbb{Z})$; this paper shows that this is the case if and only if the group $G$ admits a complete resolution and its complete cohomology is calculated via complete resolutions. It is also shown that having the periodicity isomorphisms given by the cup product with an element in $H^q (G,\mathbb{Z})$ is equivalent to silp $G$ being finite, where silp $G$ is the supremum of the injective lengths of the projective $\mathbb{Z}G$-modules. [ABSTRACT FROM AUTHOR]

Ann-category is called almost strict if its natural equivalences, except a natural equivalence of commutativity and of the distributivity, are identities. The purpose of this paper is to prove that every Ann-category is Ann-equivalent to an almost strict Ann-category of the type (R, M) and to give new interpretations of the cohomology groups H3 (R, M) of the rings R. The present paper consists, in a certain sense, of an extension of our results in [4-6]. Reading the present paper requires certain knowledge of the main results, which were announced in [3]. For completeness, we briefly recall some of the material that will be indispensable for the understanding of this paper. Throughout this paper, for the tensorial product of two objects A and B, we write AB instead of A ⊗ B, but for the morphisms we still write f ⊗ g to avoid confusion with composition. [ABSTRACT FROM AUTHOR]

We calculate the derived functors R ? * for the base change ? from the Weil-étale site to the étale site for a variety over a finite field. For smooth and proper varieties, we apply this to express Tate’s conjecture and Lichtenbaum’s conjecture on special values of ?-functions in terms of Weil-étale cohomology of the motivic complex Z( n). [ABSTRACT FROM AUTHOR]

The cohomology ring of the moduli space of stable holomorphic vector bundles of rank $n$ and degree $d$ over a Riemann surface of genus $g > 1$ has a standard set of generators when $n$ and $d$ are coprime. When $n = 2$ the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when $n > 2$ and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations. [ABSTRACT FROM AUTHOR]

We prove that the cohomological descent for proper hypercoverings holds true in rigid cohomology. [ABSTRACT FROM AUTHOR]

The article focuses on the gap cohomology group of a tower defined by Dan Talayco. It states that this group is isomorphic to the collection of gaps in the tower modulo the equivalence relation given by two gaps being equivalent if their symmetric difference is not a gap in the tower. It notes that Talayco shows that there is always some tower whose gap cohomology group has size 2.

This paper verifies a conjecture posed in a pair of papers on the fixed point sets for a class of quantum operations. Specifically, it is proved that if a quantum operation has mutually commuting operation elements that are effects forming a resolution of the identity, then the fixed point set of the quantum operation is exactly the commutant of the operation elements. [ABSTRACT FROM AUTHOR]