Related papers: Unitally nondistributive quantales
We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a $4$-element…
Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.
Let $L$ be a finite $n$-element semilattice. We prove that if $L$ has at least $127\cdot 2^{n-8}$ subsemilattices, then $L$ is planar. For $n>8$, this result is sharp since there is a non-planar semilattice with exactly $127\cdot 2^{n-8}-1$…
We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral.…
We study the finite basis problem for 4-element additively idempotent semirings whose additive reducts are semilattices of height 1. Up to isomorphism, there are 58 such algebras. We show that 49 of them are finitely based and the remaining…
We prove that the poset of $q$-decreasing words equipped with the componentwise order forms a lattice. We enumerate the join-irreducible elements for arbitrary $q>0$, and for any positive rational number $q$, we determine the number of…
A 3-dimensional quasiperiodic lattice, with overlapping unit cells and periodic in one direction, is constructed using grid and projection methods pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are the vertices of…
A (positive definite integral) quadratic form is called almost 2-universal if it represents all (positive definite integral) binary quadratic forms except those in only finitely many equivalence classes. Oh [7] determined all almost…
In this study, we introduce the concept of quasi n-absorbing elements of multiplicative lattices. A proper element q is said to be a quasi n-absorbing element of L if whenever $a^nb\le q$ implies that either $a^n\le q$ or $a^{n-1}b\le q$.…
We derive by lattice theory a universal quantum certainty relation for arbitrary $M$ observables in $N$-dimensional system, which provides a state-independent maximum lower bound on the direct-sum of the probability vectors in terms of…
Let $M$ be a cancellative and commutative monoid (written additively). The monoid $M$ is atomic if every non-invertible element can be written as a sum of irreducible elements (often called atoms in the literature). Weaker versions of…
Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of…
We explore the set of unitary matrices characterized by a given structure in the context of their applications in the field of Quantum Information. In the first part of the Thesis we focus on classification of special classes of unitary…
In residuated binars there are six non-obvious distributivity identities of $\cdot$,$/$,$\backslash$ over $\wedge, \vee$. We show that in residuated binars with distributive lattice reducts there are some dependencies among these…
We define a notion of super-transitivity for \`etale algebra objects $A \in \mathcal{C}(\mathfrak{sl}_N, k)$. This definition is a direct analogue of the notion of super-transitivity for subfactors, and measures at what depth the first…
We extend the theory of atomized semilattices to the infinite setting. We show that it is well-defined and that every semilattice is atomizable. We also study atom redundancy, focusing on complete and finitely generated semilattices and…
If a unitary transformation has a decomposition into a quantum circuit with no directed path from input $a$ to output $b$, then $a$ does not influence $b$ through the overall unitary. Conversely, it is known that if $a$ does not influence…
In 2020 Bhavale and Waphare introduced the concept of a nullity of a poset as nullity of its cover graph. In 2003 Pawar and Waphare counted all non-isomorphic lattices on n elements and n edges, which are precisely lattices of nullity one.…
We show how to decompose any density matrix of the simplest binary composite systems, whether separable or not, in terms of only product vectors. We determine for all cases the minimal number of product vectors needed for such a…
We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…