On Fuzzy Proper Exact Sequences and Fuzzy Projective Semimodules Over Semirings

As an analogue here we extend and give new horizon to semimodule theory by introducing fuzzy exact and proper exact sequences of fuzzy semi modules for generalizing well known theorems and results of semimodule theory to their fuzzy environment. We also elucidate completely the characterization of fuzzy projective semi modules via Hom functor and show that semimodule µP is fuzzy projective if and only if Hom(µP ,–) preservers the exactness of the sequence µM′ α¯−→νM β¯ −→ηM′′ with β¯ being K-regular. Some results of commutative diagram of R-semimodules having exact rows specifically the “5-lemma” to name one, were easily transferable with the novel proofs in their fuzzy context. Also, towards the end apart from the other equivalent conditions on homomorphism of fuzzy semimodules it is necessary to see that in semimodule theory every fuzzy free is fuzzy projective however the converse is true only with a specific condition.

Verifying non-isomorphism of groups

The concept of isomorphism is central to group theory, indeed to all of abstract algebra. Two groups {G, *} and {H, ο}are said to be isomorphic to each other if there exists a set bijection α from G onto H, such that $$\left( {a\;*\;b} \right)\alpha = \left( a \right)\alpha \; \circ \;(b)\alpha $$ for all a, b ∈ G. This can be illustrated by what is usually known as a commutative diagram:

Seiberg-Witten map for D-branes in large R-R field background

We obtain a Seiberg-Witten map for the gauge sector of multiple Dp-branes in a large R-R (p − 1)-form field background up to the first-order in the inverse R-R field background. By applying the Seiberg-Witten map and then electromagnetic duality on the non-commutative D3-brane theory in the large R-R 2-form background, we find the expected commutative diagram of the Seiberg-Witten map and electromagnetic duality. Extending the U(1) gauge group to the U(N) gauge group, we obtain a commutative description of the D-branes in the large R-R field background. This construction is different from the known result.

Generalized Extension in the Geometry of Goncharov Motivic and Grassmannian Configuration Chain Complexes

The purpose of this work is to propose generalized extensions of morphisms in the geometry of Grassmannian configuration and Goncharov motivic chain complexes. This work has two major divisions: in its first part the geometry of these complexes will be extended for weight n = 6, secondly, the generalization of this extension for any weight N will be presented. The generalized commutative diagram will also be exhibited

Filtrations in Module Categories, Derived Categories, and Prime Spectra

Let $R$ be a commutative noetherian ring. The notion of $n$-wide subcategories of ${\operatorname{\mathsf{Mod}}}\ R$ is introduced and studied in Matsui–Nam–Takahashi–Tri–Yen in relation to the cohomological dimension of a specialization-closed subset of ${\operatorname{Spec}}\ R$. In this paper, we introduce the notions of $n$-coherent subsets of ${\operatorname{Spec}}\ R$ and $n$-uniform subcategories of $\mathsf{D}({\operatorname{\mathsf{Mod}}}\ R)$ and explore their interactions with $n$-wide subcategories of ${\operatorname{\mathsf{Mod}}}\ R$. We obtain a commutative diagram that yields filtrations of subcategories of ${\operatorname{\mathsf{Mod}}}\ R$, $\mathsf{D}({\operatorname{\mathsf{Mod}}}\ R)$ and subsets of ${\operatorname{Spec}}\ R$ and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi, and Angeleri Hügel–Marks–Šťovíček–Takahashi–Vitória.

From noncommutative diagrams to anti-elementary classes

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form [Formula: see text]. We prove that many naturally defined classes are anti-elementary, including the following: the class of all lattices of finitely generated convex [Formula: see text]-subgroups of members of any class of [Formula: see text]-groups containing all Archimedean [Formula: see text]-groups; the class of all semilattices of finitely generated [Formula: see text]-ideals of members of any nontrivial quasivariety of [Formula: see text]-groups; the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem; the class of all semilattices of finitely generated two-sided ideals of rings; the class of all semilattices of finitely generated submodules of modules; the class of all monoids encoding the nonstable K0-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero; (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large [Formula: see text]-frame. The main underlying principle is that under quite general conditions, for a functor [Formula: see text], if there exists a noncommutative diagram [Formula: see text] of [Formula: see text], indexed by a common sort of poset called an almost join-semilattice, such that [Formula: see text] is a commutative diagram for every set [Formula: see text], [Formula: see text] for any commutative diagram [Formula: see text] in [Formula: see text], then the range of [Formula: see text] is anti-elementary.

Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra

In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving the exactness, we show that the usual three-dimensional sequence of trimmed Finite Element (FE) spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. A discussion on reconstructions of potentials and discrete [Formula: see text]-products completes the exposition.

CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA

Given a $2$-commutative diagramin a symmetric monoidal $(\infty ,2)$-category $\mathscr{E}$ where $X,Y\in \mathscr{E}$ are dualizable objects and $\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism $\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$ between the traces of $F_{X}$ and $F_{Y}$, respectively. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty ,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

Linear stability and stability of syzygy bundles

Let [Formula: see text] be a smooth irreducible projective curve and let [Formula: see text] be a complete and generated linear series on [Formula: see text]. Denote by [Formula: see text] the kernel of the evaluation map [Formula: see text]. The exact sequence [Formula: see text] fits into a commutative diagram that we call the Butler’s diagram. This diagram induces in a natural way a multiplication map on global sections [Formula: see text], where [Formula: see text] is a subspace and [Formula: see text] is the dual of a subbundle [Formula: see text]. When the subbundle [Formula: see text] is a stable bundle, we show that the map [Formula: see text] is surjective. When [Formula: see text] is a Brill–Noether general curve, we use the surjectivity of [Formula: see text] to give another proof of the semistability of [Formula: see text], moreover, we fill up a gap in some incomplete argument by Butler: With the surjectivity of [Formula: see text] we give conditions to determine the stability of [Formula: see text], and such conditions imply the well-known stability conditions for [Formula: see text] stated precisely by Butler. Finally we obtain the equivalence between the (semi)stability of [Formula: see text] and the linear (semi)stability of [Formula: see text] on [Formula: see text]-gonal curves.