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In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

In this paper, we describe an activity that involves economics and mathematics. It is included in the planning of orientation paths towards university studies within the Mathematical High School Project and is dedicated to students in the last years of high school. In particular, this research will deal with the issue of solving an economic problem using not only real analysis instruments but also geometrical topics concerning Euclidean geometry and topology. Mathematics becomes a language to understand and explain a real life problem, such as determining the optimal position of an airport, a nuclear reactor and so on. Some activities made use of dynamic geometry software and computer simulations. [ABSTRACT FROM AUTHOR]

This article provides a power series summability-based Korovkin type approximation theorem for any sequence of fuzzy positive linear operators. Using the notion of fuzzy modulus of smoothness, we also derive an associated approximation theorem concerning the fuzzy rate of convergence of these operators. Furthermore, through an example, we illustrate that our summability- based Korovin type theorem has an advantage over the fuzzy Korovkin type theorem proved in the seminal paper by Anastassiou (Stud. Univ. Babeş-Bolyai Math L(4):3–10, 2005) [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if a R + b R = e R for some nonzero idempotent e in R. Let G ′ (R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G ′ (R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G ′ (R) is ≤ 2 . Finally, we show that the graph G ′ (R) is a line graph of some graph if and only if R is either a regular ring or a local ring. [ABSTRACT FROM AUTHOR]

Solovyov (Fuzzy Sets Syst 159:2567–2585, 2008) introduced the notion of stratified Q-topological spaces, which in turn gives rise to the category Str-Q-TOP of stratified Q-topological spaces (where Q is a fixed member of a fixed variety of Ω -algebras). Singh and Srivastava (Ann. Fuzzy Math. Inform. 12:539–546, 2016) introduced the Q-topological space (Q , ⟨ { i d Q } ∪ { q ̲ ∣ q ∈ Q } ⟩) which is a Sierpinski object in the category Str-Q-TOP. In this paper, motivated by Singh (Fuzzy Sets Syst 159:2611–2615, 2008), we determine the coreflective hull of (Q , ⟨ { i d Q } ∪ { q ̲ ∣ q ∈ Q } ⟩) in the category Str-Q-TOP. We also determine the coreflective hulls of the categories Str-Dis-Q-TOP of discrete Q-topological spaces and Str-Ind-Q-TOP of stratified indiscrete Q-topological spaces, in the category Str-Q-TOP, motivated by Hoffmann (Arch Math 33:258–262, 1979) and Singh and Srivastava (Quaest. Math. 36: 167–179, 2013) [ABSTRACT FROM AUTHOR]

In a society that has to reconcile two areas which are conceptually very distant from each other, such as interdisciplinarity and specialization, the education for young people is essential to form a thought to understand complexity, thanks also to the stimuli deriving from increasingly powerful and interconnected technological and IT environments. So, it is essential to identify teaching–learning processes and methods that allow students to understand and act adequately with a constantly evolving reality. Transdisciplinary themes can be introduced to enrich the students' curriculum by anchoring them strongly to the wealth of emerging technologies, using their potential and designing educational activities focused on technology and attractive to the various spheres of knowledge. In this paper is presented an experimental activity of designing Smart objects in the Internet of Things conducted with students of the "Mathematical High School" project. [ABSTRACT FROM AUTHOR]

In this paper we study the theory Q. We prove a basic result that says that, in a sense explained in the paper, Q can be split into two parts. We prove some consequences of this result. (i) Q is not a poly-pair theory. This means that, in a strong sense, pairing cannot be defined in Q. (ii) Q does not have the Pudlák Property. This means that there two interpretations of $$\mathsf{S}^1_2$$ in Q which do not have a definably isomorphic cut. (iii) Q is not sententially equivalent with $$\mathsf{PA}^-$$ . This tells us that we cannot do much better than mutual faithful interpretability as a measure of sameness of Q and $$\mathsf{PA}^-$$ . We briefly consider the idea of characterizing Q as the minimal-in-some-sense theory of some kind modulo some equivalence relation. We show that at least one possible road towards this aim is closed. [ABSTRACT FROM AUTHOR]

The so-called non-associative MV-algebras were introduced recently by the first author and J. Kühr in order to have an appropriate tool for certain logics used in expert systems where associativity of the binary operation is excluded, see, e.g., Botur and Halaš (Arch Math Log 48:243–255, 2009). Since implication is an important logical connective in practically every propositional logic, in the present paper we investigate the implication reducts of non-associative MV-algebras. We also determine their structures based on the underlying posets. The natural question when a poset with the greatest element equipped with sectional switching involutions can be organized into an implication NMV-algebra is solved. Moreover, congruence properties of the variety of implication NMV-algebras with, respectively, without zero are investigated. Analogously to classical propositional logic, we introduce a certain kind of Sheffer operation and we obtain a one-to-one correspondence between NMV-algebras and certain algebras built up by a Sheffer-like operation together with a unary operation. [ABSTRACT FROM AUTHOR]

In this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard $$\mathsf {BL}$$ -algebras, we focus on a uniform approach to the following problem: given a left-continuous t-norm $$*$$ , find an axiomatic system (possibly with infinitary rules) which is strongly complete with respect to the standard algebra This system will be an expansion of Monoidal t-norm-based logic. First, we introduce an infinitary axiomatic system $$\mathsf {L}_*^\infty $$ , expanding the language with $$\Delta $$ and countably many truth constants, and with only one infinitary inference rule, that is inspired in Takeuti-Titani density rule. Then we show that $$\mathsf {L}_*^\infty $$ is indeed strongly complete with respect to the standard algebra . Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0, 1] satisfy some regularity conditions. [ABSTRACT FROM AUTHOR]

In this paper, we explore some unexplored consequences of Agliano et al. (Studia Logica 87:73-98, 2007) and Agliano and Montagna (J Pure Appl Algebra 181:105-129, 2003) on varieties of BL-algebras and their hoop subreducts. The main topics are: generation by finite algebras, the finite embeddability property and varieties of finite index. [ABSTRACT FROM AUTHOR]