Mahdieh Haddadi

• PhD
• Professor (Associate) at Semnan University

In this paper, we use two approaches to define the concept of fuzzy nominal sets: classic and universal algebraic. We see that the fuzzy nominal sets obtained using the universal algebraic approach (so-called fuzzy nominal sets) are within finitely supported mathematics, whereas the fuzzy nominal sets derived using the classical approach (so-called...

The purpose of this paper is to provide simple characterizations of the projective objects in the category of finitely supported M-sets. To do so, first, we introduce the notion of zero-retraction monoid and then characterize projective finitely supported M-sets where M contains a zero-retraction monoid.

The category ${\rm Rel}(\mathcal{C})$ may be formed for any category $\mathcal{C}$ with finite limits using the same objects as $\mathcal{C}$ but whose morphisms from $X$ to $Y$ are binary relations in $\mathcal{C}$, that is, subobjects of $X\times Y$. In this paper, concerning the topos ${\bf Nom}$, we study the category ${\bf Rel}({\bf Nom})$. In...

Graph theory is a useful tool to study the properties of algebraic structures. Here we are going to employ this tool to study nominal sets. To do so, we employ the freshness relation to construct certain graphs so-called fresh-graph, due to the key role of freshness relation in obtaining these graphs. We then investigate the properties of these gra...

Each nominal set 𝑋 can be equipped with a preorder relation ⪯ defined by the notion of support, so-called support-preorder. This preorder also leads us to the support topology on each nominal set. We study support-preordered nominal sets and some of their categorical properties in this paper. We also examine the topological properties of support to...

Each nominal set can be equipped with a preorder relation ⪯ defined by the notion of support, so-called support-preorder. This preorder also leads us to the support topology on each nominal set. We study support-preordered nominal sets and some of their categorical properties in this paper. We also examine the topological properties of support topo...

In this paper, similar to the Lembek's idea concerning a generalization of the notion of purity associated with a radical in the category of R-modules, we give the notion of purity related to a Hoehnke radical, d.l.i.pure, in the category of $S$-acts and investigate this notion. We also show that absolutely d.l.i.pure $S$-acts are exactly the $r$-w...

In this paper, we show that injectivity with respect to the class D of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category well-behaves. Indeed, if M is a subclass of monomorphisms, M ∩ D-injectivity well-behaves. We also introduce the notion of (r, t)-injectivity in the category S-Act, where r and t...

In this paper, we show that injectivity with respect to the class $\mathcal{D}$ of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category well-behaves. Indeed, if $\mathcal{M}$ is a subclass of monomorphisms, $\mathcal{M}\cap \mathcal{D}$-injectivity well-behaves. We also introduce the notion of $(r,t)$...

Various generalizations of the concept of injectivy, in particular injectivy with respect to a specific class of morphisms, have been intensively studied throughout the years in different categories. One of the important kinds of injectivy studied in the category R-Mod of R-modules is ${\tau}$-injectivy, for a torsion theory ${\tau}$, or in the oth...

In abelian categories like the category of R-modules and even in the category S-Act 0 of S-acts with a unique zero, idempotent radicals and torsion theories are equivalent, and the {\tau}-torsion and {\tau}- torsion free classes of a torsion theory {\tau} are closed under coproducts. These are not necessarily true in the category S-Act of S-acts. I...

Let S-Set be the category of S-sets, sets together with the actions of a semigroup S on them. And, let S-Pos be the category of S-posets, posets together with the actions compatible with the orders on them. In this paper we show that the category S-Pos is a radical extension of S-Set; that is there is a radical on the category S-Pos, the order deso...

In this paper we investigate the actions of a monoid of the form S = G∪I, where G is a group and I is an ideal of S, on sets. So, naturally, every S-act can be considered as an I¹-act. The central question here is that what is the relation between injective and weakly injective I¹-acts and injective and weakly injective S-acts? We are going to resp...

The study of monoids in the category of monoid acts leads to the notion of power action. In this paper, for a monoid T, we investigate the relationship between the category T-Act of all T-acts and the category T-Pwr of all T-power acts. For a T-power act M on a commutative monoid T, we introduce the covariant functor MM- from T-Act to T-Pwr and sho...

The notion of a Cauchy sequence in an S-poset is a useful tool to study algebraic concepts, specially the concept of injectivity. This paper is concerned with the relations between injectivity and Cauchy sequences in the category of S-posets in which S is a left zero posemi- group. We characterize subdirectly irreducible S-posets over this posemigr...

Although fuzzy set theory and sheaf theory have been developed and studied independently, Ulrich Hohle shows that a large part of fuzzy set theory is in fact a subfield of sheaf theory. Many authors have studied math- ematical structures, in particular, algebraic structures, in both categories of these generalized (multi)sets. Using Hohle's idea, w...

An algebra is called corecursive if from every coalgebra a unique coalgebra-to-algebra homomorphism exists into it. We prove that free corecursive algebras are obtained as coproducts of the terminal coalgebra (considered as an algebra) and free algebras. The monad of free corecursive algebras is proved to be the free corecursive monad, where the co...

Some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness. In this overview paper, we �rst try to refresh these smallness condition by giving the detailed proofs of the r...

Abstract. Some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness. In this overview paper, we �rst try to refresh these smallness condition by giving the detailed proof...

Sequentially dense monomorphisms were first introduced and studied by Giuli for projection algebras and followed by Ebrahimi, Mahmoudi, Moghaddasi and Shahbaz for S-acts. In this paper we use the notion of sub-Cauchy sequences and introduce the class of regular sub-sequentially dense monomorphisms for S-posets, denoted by Ms. We investigate the pro...

Abstract. Nets, useful topological tools, used to generalize certain concepts that may only be general enough in the context of metric spaces. In this work we introduce this concept in an S-poset, a poset with an action of a posemigroup S on it which is a very useful structure in computer sciences and interesting for mathematicians, and give the co...

Purity and equational compactness play a role at least in the Theories of Modules, Acts, Model, and Category. Adámek and Rosický have studied them categorically, Rothmaler model-theoretically, and some authors, including Banaschewski, Gould, and Normak have studied these notions on G-acts. We take both the group G and the set A in the definition of...

An algebra is called corecursive if from every coalgebra a unique coalgebra-to-algebra homomorphism exists into it. We prove that free corecursive algebras are obtained as a coproduct of the final coalgebra (considered as an algebra) and with free algebras. The monad of free corecursive algebras is proved to be the free corecursive monad, where the...

Nowadays purity plays a role in at least four branches of mathematics: Module Theory, Theory of Acts over semigroups, Model Theory, and Category Theory. Adámek and Rosciký have studied these notions categorically, and Rothmaler model-theoretically. Some authors including Banaschewski, Gould, and Normak have studied purity on G-acts, acts over a mon...