An algebraic approach to N-soft sets with application in decision-making using TOPSIS

In this paper, we focus on two main objectives. Firstly, we define some binary and unary operations on N-soft sets and study their algebraic properties. In unary operations, three different types of complements are studied. We prove De Morgan’s laws concerning top complements and for bottom complements for N-soft sets where N is fixed and provide a counterexample to show that De Morgan’s laws do not hold if we take different N. Then, we study different collections of N-soft sets which become idempotent commutative monoids and consequently show, that, these monoids give rise to hemirings of N-soft sets. Some of these hemirings are turned out as lattices. Finally, we show that the collection of all N-soft sets with full parameter set E and collection of all N-soft sets with parameter subset A are Stone Algebras. The second objective is to integrate the well-known technique of TOPSIS and N-soft set-based mathematical models from the real world. We discuss a hybrid model of multi-criteria decision-making combining the TOPSIS and N-soft sets and present an algorithm with implementation on the selection of the best model of laptop.

Factorization theory in commutative monoids

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

The system of sets of lengths and the elasticity of submonoids of a finite-rank free commutative monoid

Let [Formula: see text] be an atomic monoid. For [Formula: see text], let [Formula: see text] denote the set of all possible lengths of factorizations of [Formula: see text] into irreducibles. The system of sets of lengths of [Formula: see text] is the set [Formula: see text]. On the other hand, the elasticity of [Formula: see text], denoted by [Formula: see text], is the quotient [Formula: see text] and the elasticity of [Formula: see text] is the supremum of the set [Formula: see text]. The system of sets of lengths and the elasticity of [Formula: see text] both measure how far [Formula: see text] is from being half-factorial, i.e. [Formula: see text] for each [Formula: see text]. Let [Formula: see text] denote the collection comprising all submonoids of finite-rank free commutative monoids, and let [Formula: see text]. In this paper, we study the system of sets of lengths and the elasticity of monoids in [Formula: see text]. First, we construct for each [Formula: see text] a monoid in [Formula: see text] having extremal system of sets of lengths. It has been proved before that the system of sets of lengths does not characterize (up to isomorphism) monoids in [Formula: see text]. Here we use our construction to extend this result to [Formula: see text] for any [Formula: see text]. On the other hand, it has been recently conjectured that the elasticity of any monoid in [Formula: see text] is either rational or infinite. We conclude this paper by proving that this is indeed the case for monoids in [Formula: see text] and for any monoid in [Formula: see text] whose corresponding convex cone is polyhedral.

Topos points of quasi-coherent sheaves over monoid schemes

Let X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos of quasi-coherent sheaves over X. As it turns out, every topos point of is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if and are equivalent.The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ∞, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.