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H. V. Kumbhojkar - One of the best experts on this subject based on the ideXlab platform.
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Spectrum of prime L-fuzzy h-ideals of a hemiring
Fuzzy Sets and Systems, 2010Co-Authors: H. V. KumbhojkarAbstract:We redefine the concept of prime fuzzy h-ideals of a hemiring so that the fuzzy h-ideals are not necessarily 2-valued. We also introduce the concept of semiprime fuzzy h-ideals. A topological space, called the spectrum of prime fuzzy h-ideals of a commutative hemiring with unity, has been obtained. This topological space is compact and preserves isomorphisms between hemirings. The correspondence associating a hemiring with its spectrum of prime fuzzy h-ideals is shown to define a Contravariant Functor from the category of commutative hemirings with unity into the category of compact topological spaces. The spectrum of (crisp) prime h-ideals of the hemiring is a subspace which is dense in the spectrum of prime fuzzy h-ideals. Valuation lattices for all the fuzzy sets in the paper are assumed to be complete Heyting algebras.
Dario Spirito - One of the best experts on this subject based on the ideXlab platform.
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Topological properties of semigroup primes of a commutative ring
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 2017Co-Authors: Carmelo Antonio Finocchiaro, Marco Fontana, Dario SpiritoAbstract:A semigroup prime of a commutative ring R is a prime ideal of the semigroup $$(R,\cdot )$$ ( R , · ) . One of the purposes of this paper is to study, from a topological point of view, the space $${\varvec{\mathcal {S}}}(R)$$ S ( R ) of prime semigroups of R . We show that, under a natural topology introduced by B. Olberding in 2010, $${\varvec{\mathcal {S}}}(R)$$ S ( R ) is a spectral space (after Hochster), spectral extension of $${{\mathtt {Spec}}}(R)$$ Spec ( R ) , and that the assignment $$R\mapsto {\varvec{\mathcal {S}}}(R)$$ R ↦ S ( R ) induces a Contravariant Functor. We then relate—in the case R is an integral domain—the topology on $${\varvec{\mathcal {S}}}(R)$$ S ( R ) with the Zariski topology on the set of overrings of R . Furthermore, we investigate the relationship between $${\varvec{\mathcal {S}}}(R)$$ S ( R ) and the space $$\varvec{\mathcal {X}}(R)$$ X ( R ) consisting of all nonempty inverse-closed subspaces of $${{\mathtt {Spec}}}(R)$$ Spec ( R ) , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that $${\varvec{\mathcal {S}}}( R)$$ S ( R ) is a spectral retract of $$\varvec{\mathcal {X}}(R)$$ X ( R ) and we characterize when $${\varvec{\mathcal {S}}}( R)$$ S ( R ) is canonically homeomorphic to $$\varvec{\mathcal {X}}(R)$$ X ( R ) , both in general and when $${{\mathtt {Spec}}}(R)$$ Spec ( R ) is a Noetherian space. In particular, we obtain that, when R is a Bézout domain, $${\varvec{\mathcal {S}}}( R)$$ S ( R ) is canonically homeomorphic both to $$\varvec{\mathcal {X}}(R)$$ X ( R ) and to the space $$\mathtt {Overr}(R)$$ Overr ( R ) of the overrings of R (endowed with the Zariski topology). Finally, we compare the space $$\varvec{\mathcal {X}}(R)$$ X ( R ) with the space $${\varvec{\mathcal {S}}}(R(T))$$ S ( R ( T ) ) of semigroup primes of the Nagata ring R ( T ), providing a canonical spectral embedding $${\varvec{\mathcal {X}}}(R)\hookrightarrow {\varvec{\mathcal {S}}}(R(T))$$ X ( R ) ↪ S ( R ( T ) ) which makes $${\varvec{\mathcal {X}}}(R)$$ X ( R ) a spectral retract of $${\varvec{\mathcal {S}}}(R(T))$$ S ( R ( T ) ) .
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topological properties of semigroup primes of a commutative ring
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, 2017Co-Authors: Carmelo Antonio Finocchiaro, Marco Fontana, Dario SpiritoAbstract:A semigroup prime of a commutative ring R is a prime ideal of the semigroup \((R,\cdot )\). One of the purposes of this paper is to study, from a topological point of view, the space \({\varvec{\mathcal {S}}}(R)\) of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, \({\varvec{\mathcal {S}}}(R)\) is a spectral space (after Hochster), spectral extension of \({{\mathtt {Spec}}}(R)\), and that the assignment \(R\mapsto {\varvec{\mathcal {S}}}(R)\) induces a Contravariant Functor. We then relate—in the case R is an integral domain—the topology on \({\varvec{\mathcal {S}}}(R)\) with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between \({\varvec{\mathcal {S}}}(R)\) and the space \(\varvec{\mathcal {X}}(R)\) consisting of all nonempty inverse-closed subspaces of \({{\mathtt {Spec}}}(R)\), which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that \({\varvec{\mathcal {S}}}( R)\) is a spectral retract of \(\varvec{\mathcal {X}}(R)\) and we characterize when \({\varvec{\mathcal {S}}}( R)\) is canonically homeomorphic to \(\varvec{\mathcal {X}}(R)\), both in general and when \({{\mathtt {Spec}}}(R)\) is a Noetherian space. In particular, we obtain that, when R is a Bezout domain, \({\varvec{\mathcal {S}}}( R)\) is canonically homeomorphic both to \(\varvec{\mathcal {X}}(R)\) and to the space \(\mathtt {Overr}(R)\) of the overrings of R (endowed with the Zariski topology). Finally, we compare the space \(\varvec{\mathcal {X}}(R)\) with the space \({\varvec{\mathcal {S}}}(R(T))\) of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding \({\varvec{\mathcal {X}}}(R)\hookrightarrow {\varvec{\mathcal {S}}}(R(T))\) which makes \({\varvec{\mathcal {X}}}(R)\) a spectral retract of \({\varvec{\mathcal {S}}}(R(T))\).
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topological properties of semigroup primes of a commutative ring
arXiv: Commutative Algebra, 2017Co-Authors: Carmelo Antonio Finocchiaro, Marco Fontana, Dario SpiritoAbstract:A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space $\scal(R)$ of prime semigroups of $R$. We show that, under a natural topology introduced by B. Olberding in 2010, $\scal(R)$ is a spectral space (after Hochster), spectral extension of $\Spec(R)$, and that the assignment $R\mapsto\scal(R)$ induces a Contravariant Functor. We then relate -- in the case $R$ is an integral domain -- the topology on $\scal(R)$ with the Zariski topology on the set of overrings of $R$. Furthermore, we investigate the relationship between $\scal(R)$ and the space $\boldsymbol{\mathcal{X}}(R)$ consisting of all nonempty inverse-closed subspaces of $\spec(R)$, which has been introduced and studied in C.A. Finocchiaro, M. Fontana and D. Spirito, "The space of inverse-closed subsets of a spectral space is spectral" (submitted). In this context, we show that $\scal( R)$ is a spectral retract of $\boldsymbol{\mathcal{X}}(R)$ and we characterize when $\scal( R)$ is canonically homeomorphic to $\boldsymbol{\mathcal{X}}(R)$, both in general and when $\spec(R)$ is a Noetherian space. In particular, we obtain that, when $R$ is a Bezout domain, $\scal( R)$ is canonically homeomorphic both to $\boldsymbol{\mathcal{X}}(R)$ and to the space $\overr(R)$ of the overrings of $R$ (endowed with the Zariski topology). Finally, we compare the space $\boldsymbol{\mathcal{X}}(R)$ with the space $\scal(R(T))$ of semigroup primes of the Nagata ring $R(T)$, providing a canonical spectral embedding $\xcal(R)\hookrightarrow\scal(R(T))$ which makes $\xcal(R)$ a spectral retract of $\scal(R(T))$.
Spirito Dario - One of the best experts on this subject based on the ideXlab platform.
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Topological properties of semigroup primes of a commutative ring
'Springer Science and Business Media LLC', 2017Co-Authors: Finocchiaro, Carmelo A., Fontana Marco, Spirito DarioAbstract:A semigroup prime of a commutative ring R is a prime ideal of the semigroup (R, ·). One of the purposes of this paper is to study, from a topological point of view, the space S(R) of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, S(R) is a spectral space (after Hochster), spectral extension of Spec(R) , and that the assignment Râ\u86¦ S(R) induces a Contravariant Functor. We then relateâ\u80\u94in the case R is an integral domainâ\u80\u94the topology on S(R) with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between S(R) and the space X(R) consisting of all nonempty inverse-closed subspaces of Spec(R) , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that S(R) is a spectral retract of X(R) and we characterize when S(R) is canonically homeomorphic to X(R) , both in general and when Spec(R) is a Noetherian space. In particular, we obtain that, when R is a Bézout domain, S(R) is canonically homeomorphic both to X(R) and to the space Overr(R) of the overrings of R (endowed with the Zariski topology). Finally, we compare the space X(R) with the space S(R(T)) of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding X(R) â\u86ª S(R(T)) which makes X(R) a spectral retract of S(R(T))
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Topological properties of semigroup primes of a commutative ring
2017Co-Authors: Finocchiaro, Carmelo A., Fontana Marco, Spirito DarioAbstract:A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space $\scal(R)$ of prime semigroups of $R$. We show that, under a natural topology introduced by B. Olberding in 2010, $\scal(R)$ is a spectral space (after Hochster), spectral extension of $\Spec(R)$, and that the assignment $R\mapsto\scal(R)$ induces a Contravariant Functor. We then relate -- in the case $R$ is an integral domain -- the topology on $\scal(R)$ with the Zariski topology on the set of overrings of $R$. Furthermore, we investigate the relationship between $\scal(R)$ and the space $\boldsymbol{\mathcal{X}}(R)$ consisting of all nonempty inverse-closed subspaces of $\spec(R)$, which has been introduced and studied in C.A. Finocchiaro, M. Fontana and D. Spirito, "The space of inverse-closed subsets of a spectral space is spectral" (submitted). In this context, we show that $\scal( R)$ is a spectral retract of $\boldsymbol{\mathcal{X}}(R)$ and we characterize when $\scal( R)$ is canonically homeomorphic to $\boldsymbol{\mathcal{X}}(R)$, both in general and when $\spec(R)$ is a Noetherian space. In particular, we obtain that, when $R$ is a B\'ezout domain, $\scal( R)$ is canonically homeomorphic both to $\boldsymbol{\mathcal{X}}(R)$ and to the space $\overr(R)$ of the overrings of $R$ (endowed with the Zariski topology). Finally, we compare the space $\boldsymbol{\mathcal{X}}(R)$ with the space $\scal(R(T))$ of semigroup primes of the Nagata ring $R(T)$, providing a canonical spectral embedding $\xcal(R)\hookrightarrow\scal(R(T))$ which makes $\xcal(R)$ a spectral retract of $\scal(R(T))$.Comment: 21 page
Inna Entova Aizenbud - One of the best experts on this subject based on the ideXlab platform.
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Schur Weyl duality in complex rank
2016Co-Authors: Inna Entova AizenbudAbstract:This thesis gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1) we give a definition of a complex tensor power of V. This is an Ind-object of the Deligne category Rep(St) equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(St) and a localization of the category Op/t,v (the parabolic category 0 for gl(V) associated with the pair (V, 1)). In particular, we obtain an exact Contravariant Functor SWt from the category Repab(St) (the abelian envelope of the category Rep(St)) to a certain quotient of the category Op/t v. This quotient, denoted by 0 p/t v, is obtained by taking the full subcategory of Op/t v consisting of modules of degree t, and localizing by the subcategory of finite dimensional modules. It turns out that the Contravariant Functor SWt makes Op/t v a Serre quotient of the category Repab(St)OP, and the kernel of SWt can be explicitly described. In the second part of this thesis, we consider the case when V = C[infinity] . We define the appropriate version of the parabolic category 0 and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories Op/t1CN with N tending to infinity. The Schur-Weyl Functors SWt,CN then give an anti-equivalence between the category Op[infinity]/t C[infinity]and the category Repab(Se). This duality provides an unexpected tensor structure on the category Op[infinity]/t C[infinity].
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Schur-Weyl duality for Deligne categories
arXiv: Representation Theory, 2014Co-Authors: Inna Entova AizenbudAbstract:This paper gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1), we give a definition of a complex tensor power of V. This is an Ind-object of the Deligne category Rep(S_t), equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(S_t) and a localization of the parabolic category O for gl(V) associated with the pair (V, 1). In particular, we obtain an exact Contravariant Functor SW from the category Rep^{ab}(S_t) (the abelian envelope of the category Rep(S_t)) to a certain quotient \hat{O} of the parabolic category O. This quotient is obtained by taking the full subcategory consisting of modules of degree t, and localizing by the subcategory of finite dimensional modules. It turns out that the Contravariant Functor SW makes \hat{O} a Serre quotient of the category Rep^{ab}(S_t)^{op}, and the kernel of SW can be explicitly described.
Entova Aizenbud Inna - One of the best experts on this subject based on the ideXlab platform.
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Schur Weyl duality in complex rank
Massachusetts Institute of Technology, 2016Co-Authors: Entova Aizenbud InnaAbstract:Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.Cataloged from PDF version of thesis.Includes bibliographical references (pages 207-208).This thesis gives an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space V (i.e. a vector space V with a distinguished non-zero vector 1) we give a definition of a complex tensor power of V. This is an Ind-object of the Deligne category Rep(St) equipped with a natural action of gl(V). This construction allows us to describe a duality between the abelian envelope of the category Rep(St) and a localization of the category Op/t,v (the parabolic category 0 for gl(V) associated with the pair (V, 1)). In particular, we obtain an exact Contravariant Functor SWt from the category Repab(St) (the abelian envelope of the category Rep(St)) to a certain quotient of the category Op/t v. This quotient, denoted by 0 p/t v, is obtained by taking the full subcategory of Op/t v consisting of modules of degree t, and localizing by the subcategory of finite dimensional modules. It turns out that the Contravariant Functor SWt makes Op/t v a Serre quotient of the category Repab(St)OP, and the kernel of SWt can be explicitly described. In the second part of this thesis, we consider the case when V = C[infinity] . We define the appropriate version of the parabolic category 0 and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories Op/t1CN with N tending to infinity. The Schur-Weyl Functors SWt,CN then give an anti-equivalence between the category Op[infinity]/t C[infinity]and the category Repab(Se). This duality provides an unexpected tensor structure on the category Op[infinity]/t C[infinity].by Inna Entova Aizenbud.Ph. D
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Schur-Weyl duality for Deligne categories II: the limit case
'Mathematical Sciences Publishers', 2015Co-Authors: Entova Aizenbud InnaAbstract:This paper is a continuation of a previous paper of the author, which gave an analogue to the classical Schur-Weyl duality in the setting of Deligne categories. Given a finite-dimensional unital vector space $V$ (a vector space $V$ with a chosen non-zero vector $\mathbf{1}$), we constructed a complex tensor power of $V$: an $Ind$-object of the Deligne category $\underline{Rep}(S_t)$ which is a Harish-Chandra module for the pair $(\mathfrak{gl}(V), \bar{\mathfrak{P}}_{\mathbf{1}})$, where $\bar{\mathfrak{P}}_{\mathbf{1}} \subset GL(V)$ is the mirabolic subgroup preserving the vector $\mathbf{1}$. This construction allowed us to obtain an exact Contravariant Functor $\widehat{SW}_{t, V}$ from the category $\underline{Rep}^{ab}(S_t)$ (the abelian envelope of the category $\underline{Rep}(S_t)$) to a certain localization of the parabolic category $\mathcal{O}$ associated with the pair $(\mathfrak{gl}(V), \bar{\mathfrak{P}}_{\mathbf{1}})$. In this paper, we consider the case when $V = \mathbb{C}^{\infty}$. We define the appropriate version of the parabolic category $\mathcal{O}$ and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories $\widehat{\mathcal{O}}^{\mathfrak{p}}_{t,\mathbb{C}^N}$ with $N$ tending to infinity. The Schur-Weyl Functors $\widehat{SW}_{t, \mathbb{C}^N}$ then give an anti-equivalence between this category and the category $\underline{Rep}^{ab}(S_t)$. This duality provides an unexpected tensor structure on the category $\widehat{\mathcal{O}}^{\mathfrak{p}_{\infty}}_{t, \mathbb{C}^{\infty}}$.Comment: Continuation of arXiv:1403.5509 [math.RT]. 24 pages. Comments welcom