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We give a definition of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification of a quantum cluster algebra,…
We address the formation and propagation of multi-spot soliton packets in saturable Kerr nonlinear media with an imprinted harmonic transverse modulation of the refractive index. We show that, in sharp contrast to homogeneous media where…
For affine monoids of dimension 2 with embedding dimension 2 and 3, we study the problem of determining when a vector is an element of the monoid, and the problem of determining the elasticity of a monoid element.
We show that if G is a finite group then no chain of modular elements in its subgroup lattice L(G) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L(G) has length at least two…
We identify all semidualizing modules over certain classes of ladder determinantal rings over a field ${\mathsf k}$. Specifically, given a ladder of variables $Y$, we show that the ring ${\mathsf k}[Y]/I_t(Y)$ has only trivial semidualizing…
We show that the category of log homotopy types is a full subcategory of a category of homotopy types with modulus.
In this paper we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The…
A notion of {\em normal submonoid} of a monoid $M$ is introduced that generalizes the normal subgroups of a group. When ordered by inclusion, the set $\mathsf{NorSub}(M)$ of normal submonoids of $M$ is a complete lattice. Joins are…
In 2002 Thakare et al.\ counted non-isomorphic lattices on $n$ elements, having nullity up to two. In 2020 Bhavale and Waphare introduced the concept of RC-lattices as the class of all lattices in which all the reducible elements are…
A join-semilattice $L$ is said to be conjunctive if it has a top element $1$ and it satisfies the following first-order condition: for any two distinct $a,b\in L$, there is $c\in L$ such that either $a\vee c\not=1=b\vee c$ or $a\vee…
We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower…
An element e of an ordered semigroup $(S,\cdot,\leq)$ is called an ordered idempotent if $e\leq e^2$. We call an ordered semigroup $S$ idempotent ordered semigroup if every element of $S$ is an ordered idempotent. Every idempotent semigroup…
Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of…
We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin…
In this paper, we study the posets of classes of subgroups of finite group having same set of orders of elements. We show that this poset is a chain only in the case of p-groups and moreover, we characterize all finite groups for which this…
The aim of this paper is sketch a theory of divisibility and factorisation in topological monoids, where finite products are replaced by convergent products. The algebraic case can then be viewed as the special case of discretely…
We show that if $(M,\tensor,I)$ is a monoidal model category then $\REnd_M(I)$ is a (weak) 2-monoid in $\sSet$. This applies in particular when $M$ is the category of $A$-bimodules over a simplicial monoid $A$: the derived endomorphisms of…
Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms…
We classify all matroids with at most 8 elements that have the half-plane property, and we provide a list of some matroids on 9 elements that have, and that do not have the half-plane property. Furthermore, we prove that several classes of…
The homotopy type of the complement of a complex coordinate subspace arrangement is studied by fathoming out the connection between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy…