A Certain Structure of Bipolar Fuzzy Subrings

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define (α,β)-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.

Negacyclic codes of prime power length over the finite non-commutative chain ring 𝔽pm[u,𝜃] 〈u2〉

In this paper, we focus on the algebraic structure of left negacyclic codes of length [Formula: see text] over the finite non-commutative chain ring [Formula: see text] where [Formula: see text] is an automorphism on [Formula: see text]. After that, the number of codewords of all left negacyclic codes is obtained. For each left negacyclic code, we also obtain the structure of its right dual code. In the remaining result, the number of distinct left negacyclic codes is given. Finally, a one-to-one correspondence between left negacyclic and left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] is constructed via ring isomorphism, which carries over the results regarding left negacyclic codes corresponding to left [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] where [Formula: see text] is a nonzero element of the field [Formula: see text] such that [Formula: see text].

Mappings preserving sum of products a∘b+ba∗ on factor von Neumann algebras

Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ:A→B satisfies Φ(a∘b+ba∗)=Φ(a)∘Φ(b)+Φ(b)Φ(a)∗ (where ∘ is the special Jordan product on A and B, respectively), for all elements a,b∈A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism.

Borel Localization for a Circle Action

This chapter explores Borel localization for a circle action. For a circle action, the Borel localization theorem says that up to torsion, the equivariant cohomology of an S1-manifold is concentrated on its fixed point set and that the isomorphism in localized equivariant cohomology of the manifold and its fixed point set is a ring isomorphism. This is clearly an important result in its own right. Moreover, since the fixed point set is a regular submanifold and is usually simpler than the manifold, the Borel localization theorem sometimes allows one to obtain the ring structure of the equivariant cohomology of an S1-manifold from that of its fixed point set. The chapter demonstrates this method with the example of S1 acting on S2 by rotations.

Strong cohomological rigidity of toric varieties

Every cohomology ring isomorphism between two non-singular complete toric varieties (respectively, two quasitoric manifolds), with second Betti number 2, is realizable by a diffeomorphism (respectively, homeomorphism).

Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies {C^{*}}-isomorphism of {C^{*}}-algebras for graphs E and F in which every cycle has an exit.

Classification of Bott Manifolds up to Dimension 8

We show that three- and four-stage Bott manifolds are classified up to diffeomorphism by their integral cohomology rings. In addition, any cohomology ring isomorphism between two three-stage Bott manifolds can be realized by a diffeomorphism between the Bott manifolds.