ABSTRACT In this paper, we propose the strong* proximity lattices, and prove that the categories and are categorically equivalent, building on the framework established by Achim Jung and Philipp Sünderhauf. denotes the category of all QFS‐domains and continuous functions. denotes the category of the strong* proximity lattices with the approximable relations. This answers the question posed by Achim Jung at the conference ISDT'13.
We introduce an -ary operation on the class of the ordinals, which is strictly monotone in all significant cases. We provide order-theoretical characterizations as the rank of a sequence in a well-founded order and as a mixed sum of the ordinals in the sequence.
In this study, we extend the additively generated triangular norms from the framework of the unit interval to that of partially ordered sets. We present several conditions under which this formula yields a t-norm, where are partially ordered sets, and are monotone functions and is a t-norm/t-conorm on . The partially ordered semigroups induced by are order-preserving/order-reversing homomorphic to semigroup deformations of the semigroup induced by .
Automated deduction based on contradiction separation extends the binary resolution principle, offering a novel approach to deductive inference rules. Constructing standard contradictions is essential for its efficiency. This paper systematically investigates two new types of standard contradictions in propositional and first-order logic, enriching the library of standard contradictions and enhancing its effectiveness. First, we define two types of standard contradictions: sign-boundary contradictions and diagonal vacancy-type contradictions. Next, we propose the corresponding construction methods and present their properties related to contradiction composition and literal addition. Furthermore, we explore the transformations between these two types of contradictions and analyze the conditions necessary to construct standard contradictions. Finally, we extend these findings to first-order logic, demonstrating their applicability in more complex logical systems.
Here, we investigate congruence regularity and congruence coherence in Ockham algebras with balanced pseudocomplementation. We show that such an algebra is congruence regular if and only if it is congruence coherent, which is the case if and only if for each congruence on , the smallest congruence that collapses is equal to , this in turn is equivalent to that is a trivalent & Lstrok;ukasiewicz algebra.
This article introduces a novel definition of -Hilfer fractional derivative. Based on this derivative, some fractional Wirtinger type inequalities are established for the spaces, where by using H & ouml;lder's inequality. Various related special cases are also presented. To validate our main results, examples with graphical representations are provided. Applications of -Hilfer fractional Wirtinger-type inequalities are demonstrated in terms of arithmetic mean and geometric mean-type inequality.
We introduce the concept of right-continuous mappings from [0,1) to the power set of finite elements in a complete semilattice and define a closure operator on the set of right-continuous mappings. We show that the lattice of fuzzy ideals on the semilattice of finite elements is isomorphic to the lattice of closed elements concerning the closure operator. Furthermore, we give equivalent conditions for regular closure operators on the set of right-continuous mappings to be quasi-algebraic. Finally, we prove the lattice of fuzzy ideals on the set of finite elements in an algebraic semilattice is subdirectly embedded into the product of copies of the semilattice, and show the subdirect embedding has a universal mapping property.
In this study, the Minkowski and Fej & eacute;r-Hermite-Hadamard (H-H) type inequalities are generalized by utilizing the modified Atangana-Baleanu (A-B) fractional operators. These fractional operators, defined by their nonlocal and nonsingular kernels provide a new way to generalize these classical inequalities. The inequalities are verified through several illustrative examples and corresponding graphs. A new application involving the digamma function is presented to demonstrate the significance of the results. This research opens new avenues for establishing further inequalities via fractional operators.
The classical approach to establishing the existence of solutions for implicit fractional differential equations (FDEs) relies on non-constructive fixed-point theorems, notably Banach's contraction principle. This paper provides a foundational analysis of such an existence theorem from the perspectives of proof theory and reverse mathematics. We focus on a model problem involving an implicit FDE of order with -Caputo derivative and anti-periodic boundary conditions. First, by applying methods of proof mining to the standard contraction mapping argument, we extract an explicit, computable rate of convergence for the associated Picard iteration. This yields a quantitative refinement of the classical existence proof and provides an a priori error estimate for numerical approximations. Second, we analyze the set-existence axioms required for the proof within the framework of subsystems of second-order arithmetic. We show that the existence theorem is provable in the system (Arithmetical Comprehension Axiom), establishing an upper bound on its logical strength. Furthermore, we provide a detailed analysis indicating that the theorem has logical strength at least that of , placing it within the level of the reverse mathematics hierarchy. Our work bridges fractional calculus and mathematical logic, providing both constructive insights for numerical analysis and a clear logical classification of a central result in the theory of FDEs.
This study introduces a novel fuzzy algebraic structure, termed "convex ordered fuzzy subrings" (COF SRs), within the framework of "ordered fuzzy rings" (OFRs). We provide a rigorous formalization of L-subrings, L-ideals, and L-convex substructures, where L denotes a complete lattice equipped with a binary, order-compatible t-norm, integrating concepts from convexity theory, fuzzy set theory, and Heyting algebra. We investigate the preservation of convexity and order under ordered fuzzy ring homomorphisms (OFRHs) and their kernels, focusing on convex L-subrings that are closed under fuzzy addition, multiplication, and additive inverses. These substructures induce L-convex structures with closure properties under Cartesian products and infima, and their behavior under homomorphisms highlights their algebraic and topological significance. Using an axiomatic framework, we establish conditions on lattice-valued binary operations necessary to maintain convexity in fuzzy substructures. Illustrative examples, structural analyses, and interconnections are provided to substantiate these results. This work addresses existing theoretical gaps in fuzzification, ordering, convexity, and homomorphic mappings, offering a unified foundation for the study of OFRs with potential applications in algebraic analysis, logic, information theory, and soft computing.
The use of nonstandard methods to characterize properties of weak, strong and mixed extensions of congruences to ultrafilters has been the main topic of several recent papers, focused mostly on congruences and divisions. We show that similar methods can be used to extend these characterizations to arbitrary relations and their interplay.
The aim of this paper is to give natural examples of $\mathbf{\Sigma}_1^1$-complete and $\mathbf{\Pi}_1^1$-complete sets. In the first part, we consider ideals on $\omega$. In particular, we show that the Hindman ideal $\mathcal{H}$ is $\mathbf{\Pi}_1^1$-complete and consider a number of ideals generated in the similar fashion. Moreover, we show that the ideal $\mathcal{D}$ is also $\mathbf{\Pi}_1^1$-complete. In the second part, we focus on families of trees (on $\omega$ and $2$). We show that the family of trees containing Silver trees in $\mathbf{\Sigma}_1^1$-complete.
Hoover and Keisler's Probability Logic is a variant of first-order logic that replaces the standard quantifiers V and there exists with probability quantifiers (Px >= r). The expression (Px >= r)Phi(x) is interpreted as "the set {x: Phi(x)} has probability at least r." We extend this logic to capacities, a generalization of measures that need not be additive, allowing statements of the form (fx >= r)Phi(x), interpreted as "the set {x: Phi(x)) has capacity at least r." By axiomatizing the defining properties of a special class of capacities called strongly subadditive capacities, we establish a completeness theorem for our logic with capacity quantifiers.
This work employs the techniques to present novel structures to diamond-proportional to Hardy-type inequalities with boundedness. To this end, the properties of sub-multiplicative convex functions, H & ouml;lder's inequality, Jensen's inequality, and the chain rule are utilized. Additionally, the findings of this study are integrated with those of time-scale calculus and extended. The results sometimes yield constant-valued counterparts to some inequalities identified in the existing literature.
We show that for quasivarieties of p‐algebras the properties of (i) having decidable first‐order theory and (ii) having decidable first‐order theory of the finite members, coincide. The only two quasivarieties with these properties are the trivial variety and the variety of Boolean algebras. This contrasts sharply, even for varieties, with the situation in Heyting algebras where decidable varieties do not coincide with finitely decidable ones.
We propose the notions of uniform local weak o-minimality and *-local weak o-minimality. Local monotonicity theorems hold in definably complete locally o-minimal structures and uniformly locally o-minimal structures of the second kind. In this paper, we demonstrate new local monotonicity theorems for uniformly locally weakly o-minimal structures of the second kind and for locally o-minimal structures under the assumption called the univariate *-continuity property. We also prove that several formulas for dimension of definable sets which hold in definably complete locally o-minimal structures also hold in *-locally weakly o-minimal structures possessing the univariate *-continuity property.
This paper explores the coloring problem, focusing on the existence of uniformly colored substructures. The study primarily examines random graphs with edge coloring and their generic substructures. The key finding is that the absence of a monochromatic generic substructure corresponds to increased instability, meaning that the colored random graph hereditarily possesses the strict order property.
We provide, for each natural number n and each class among D-n(Sigma(0)(1)), D-n(Sigma(0)(1)), D2n+1(Sigma(0)(1))circle plus D2n+1(Sigma(0)(1)), a regular language whose associated omega-power is complete for this class.
In this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality m$\mathfrak {m}$, denoted by I(m)$I(\mathfrak {m})$, J(m)$J(\mathfrak {m})$ and mm$\mathfrak {m}<^>\mathfrak {m}$, respectively. Among our results, we show that "seq1-1(m)not equal I(m)not equal seq(m)$\operatorname{seq}<^>{1-1}(\mathfrak {m})\ne I(\mathfrak {m})\ne \operatorname{seq}(\mathfrak {m})$", "seq1-1(m)not equal J(m)not equal seq(m)$\operatorname{seq}<^>{1-1}(\mathfrak {m})\ne J(\mathfrak {m})\ne \operatorname{seq}(\mathfrak {m})$" and "seq1-1(m){1-1}(\mathfrak {m})<\mathfrak {m}<^>\mathfrak {m}\ne \operatorname{seq}(\mathfrak {m})$" are provable for an arbitrary infinite cardinal m$\mathfrak {m}$, and these are the best possible results, in the Zermelo-Fraenkel set theory (ZF$\mathsf {ZF}$) without the Axiom of Choice. Also, we show that it is relatively consistent with ZF$\mathsf {ZF}$ that there exists an infinite cardinal m$\mathfrak {m}$ such that S(m)=I(m)
