Algebraic Properties of Intuitionistic L-fuzzy Multiset Finite Automata

Algrbraic properties and structures of intuitionistic L -fuzzy multiset finite automata (ILFMA) are discussed through congruences on a semigroup in this paper. Firstly,we put forward the notion of the intuitionistic L -fuzzy compatible relation, the compatible monoid associated to the intuitionistic L- fuzzy compatible relation can be effectively constructed, and we construct two finite monoids through two different congruence relations on a given ILFMA, then we also prove that they are isomorphic. Furthermore, using the quotient structure of ILFMA, algebraic properties of intuitionistic L -fuzzy multiset transformation semigroup are discussed. According to intuitionistic L -admissible relation and homomorphism of ILFMA, we show that there is a bijective correspondence between an ILFMA and the quotient structure of another ILFMA.

Mass Gap Problem Solution in the Superfluid Quantum Space Model

A given problem in physics can be solved if it is well formulated. Well formulated means that it has a bijective correspondence to physical reality. Mass Gap Problem has no bijective correspondence with the physical reality and is that’s why not solvable mathematically. It can be solved in the frame of quantum mechanics by the formulation of the photon’s mass accordingly to the Planck-Einstein relation.

Covering classes and 1-tilting cotorsion pairs over commutative rings

We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} provides for covers, that is when the class 𝒜 {\mathcal{A}} is a covering class. We use Hrbek’s bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni–Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if 𝒢 {\mathcal{G}} is the Gabriel topology associated to the 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} , and R 𝒢 {R_{\mathcal{G}}} is the ring of quotients with respect to 𝒢 {\mathcal{G}} , we show that if 𝒜 {\mathcal{A}} is covering, then 𝒢 {\mathcal{G}} is a perfect localisation (in Stenström’s sense [B. Stenström, Rings of Quotients, Springer, New York, 1975]) and the localisation R 𝒢 {R_{\mathcal{G}}} has projective dimension at most one as an R-module. Moreover, we show that 𝒜 {\mathcal{A}} is covering if and only if both the localisation R 𝒢 {R_{\mathcal{G}}} and the quotient rings R / J {R/J} are perfect rings for every J ∈ 𝒢 {J\in\mathcal{G}} . Rings satisfying the latter two conditions are called 𝒢 {\mathcal{G}} -almost perfect.

Mass Gap Problem Physical Solution

A given problem in physics can be solved if it is well formulated. Well formulated means that it has a bijective correspondence to physical reality. Mass Gap Problem has no bijective correspondence with the physical reality and is that’s why not solvable mathematically. It can be solved physically by the formulation of the photon’s mass accordingly to the Planck-Einstein relation.

Mass Gap Problem Physical Solution

A given problem in physics can be solved if it is well formulated. Well formulated means that it has a bijective correspondence to physical reality. Mass Gap Problem has no bijective correspondence with the physical reality and is that’s why not solvable mathematically. It can be solved physically by the formulation of the photon’s mass accordingly to the Planck-Einstein relation.

Coarse compactifications and controlled products

We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

L-fuzzifying antimatroids: A fuzzy approach to the generalization of shelling precedence structures

Crisp antimatroid is a combinatorial abstraction of convexity. It also can be incorporated into the greedy algorithm in order to seek the optimal solutions. Nevertheless, this kind of significant classical structure has inherent limitations in addressing fuzzy optimization problems and abstracting fuzzy convexities. This paper introduces the concept of L-fuzzifying antimatroid associated with an L-fuzzifying family of feasible sets. Several relevant fundamental properties are obtained. We also propose the concept of L-fuzzifying rank functions for L-fuzzifying antimatroids, and then investigate their axiomatic characterizations. Finally, we shed light upon the bijective correspondence between an L-fuzzifying antimatroid and its L-fuzzifying rank function.

The mysterious story of square ice, piles of cubes, and bijections

When combinatorialists discover two different types of objects that are counted by the same numbers, they usually want to prove this by constructing an explicit bijective correspondence. Such proofs frequently reveal many more details about the relation between the two types of objects than just equinumerosity. A famous set of problems that has resisted various attempts to find bijective proofs for almost 40 y is concerned with alternating sign matrices (which are equivalent to a well-known physics model for two-dimensional ice) and their relations to certain classes of plane partitions. In this paper we tell the story of how the bijections were found.

Decomposition and classification of length functions

We study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

On Bijective Correspondence between Fuzzy Reflexive Approximation Spaces and Fuzzy Transformation Systems

The objective of this paper is to establish the relationship between fuzzy approximation operators and fuzzy transformation systems. We show that for each upper fuzzy transformation system there exists a fuzzy reflexive approximation space and vice-versa. We further establish such relationship between lower fuzzy transformation systems and fuzzy reflexive approximation spaces under the condition that the underline lattice structure satisfies double negation law.