In this paper, we study the concept of $mathcal C$-reticulation for the category $mathcal C$ whose objects are lattice-valued maps. The relation between the free objects in $mathcal C$ and the $mathcal C$-reticulation of rings and modules is discussed. Also, a method to construct $mathcal C$-reticulation is presented, in the case where $mathcal C$ is equational. Some relations between the conce...

Let $E$ be a finite set and $\mathcal P$, $\mathcal S$, $\mathcal L$ three classes of subsets of $E$, and $r$ a function defined on $2^E$. In this paper, we give an algorithm for testing if the quadruple $(\mathcal P, \mathcal S, \mathcal L, r)$ is the locked structure of a given matroid, i.e., recognizing if $(\mathcal P, \mathcal S, \mathcal L, r)$ defines a matroid. This problem is intractab...

In this note we deal with intuitionistic modal logics over $\mathcal{M}\mathcal{I}PC$ and predicate superintuitionistic logics. We study the correspondence between the lattice of all (normal) extensions of MTPC and the lattice of all predicate superintuitionistic logics. Let $\mathrm{L}_{Prop}$ denote a propositional language which contains two modal operators $\square$ and $\mathrm{O}$ , and $...

In this article, we develop a notion of Quillen bifibration which combines the two notions of Grothendieck bifibration and of Quillen model structure. In particular, given a bifibration $p:\mathcal E\to\mathcal B$, we describe when a family of model structures on the fibers $\mathcal E_A$ and on the basis category $\mathcal B$ combines into a model structure on the total category $\mathcal E$, ...

The subalgebra membership problem is the problem of deciding if a given element belongs to an algebra given by a set of generators. This is one of the best established computational problems in algebra. We consider a variant of this problem, which is motivated by recent progress in the Constraint Satisfaction Problem, and is often referred to as the Subpower Membership Problem (SMP). In the SMP...

We introduce a family of mathematical objects called $\mathcal{P}$-schemes, where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A $\mathcal{P}$-scheme is a collection of partitions of the right coset spaces $H\backslash G$, indexed by $H\in\mathcal{P}$, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as the notion of $...

In this paper, we introduce $(mathcal{C},mathcal{C}')$-controlled continuous $g$-Bessel families and their multipliers in Hilbert spaces and investigate some of their properties. We show that under some conditions sum of two $(mathcal{C},mathcal{C}')$-controlled continuous $g$-frames is a $(mathcal{C},mathcal{C}')$-controlled continuous $g$-frame. Also, we investigate when a $(mathcal{C},mathca...

Phaseless reconstruction from space-time samples is a nonlinear problem of recovering a function $x$ in a Hilbert space $\mathcal{H}$ from the modulus of linear measurements $\{\lvert \langle x, \phi_i\rangle \rvert$, $ \ldots$, $\lvert \langle A^{L_i}x, \phi_i \rangle \rvert : i \in\mathscr I\}$, where $\{\phi_i; i \in\mathscr I\}\subset \mathcal{H}$ is a set of functionals on $\mathcal{H}$, a...

Let $\Gamma$ be a $d$-bounded distance-regular graph with $d\geq 3$. Suppose that $P(x)$ is a set of all strongly closed subgraphs containing $x$ and that $P(x,i)$ is a subset of $P(x)$ consisting of the elements of $P(x)$ with diameter $i$. Let ${\mathcal{L}}'(x,i)$ be the set generated by the join of the elements in $P(x,i)$. By ordering ${\mathcal{L}}'(x,i)$ by inclusion or reverse inclusion...