Related papers: Bioperational Multisets in Various Semi-rings
Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with…
Motivated by appearance of multisemigroups in the study of additive $2$-categories, we define and investigate the notion of a multisemigroup with multiplicities. This notion seems to be better suitable for applications in higher…
Multisets are sets that allow repetition of elements. As such, multisets pave the way to a number of interesting possibilities of theoretical and applied nature. In the present work, after revising the main aspects of traditional sets, we…
The semiring of discrete dynamical systems is a simple algebraic model for modularity in deterministic systems. The objects of the semiring are finite transformations (viewed as directed graphs and regarded up to isomorphism), the sum of…
The present work looks at semiautomatic rings with automatic addition and comparisons which are dense subrings of the real numbers and asks how these can be used to represent geometric objects such that certain operations and…
We introduce the notion of a bicocycle double cross product (resp. sum) Lie group (resp. Lie algebra), and a bicocycle double cross product bialgebra, generalizing the unified products. On the level of Lie groups the construction yields a…
Multisets are an intuitive extension of the traditional concept of sets that allow repetition of elements, with the number of times each element appears being understood as the respective multiplicity. Recent generalizations of multisets to…
For several semirings S, two weighted finite automata with multiplicities in S are equivalent if and only if they can be connected by a chain of simulations. Such a semiring S is called "proper". It is known that the Boolean semiring, the…
We investigate the class of bisymmetric and quasitrivial binary operations on a given set $X$ and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations.…
Binary multirelations generalise binary relations by associating elements of a set to its subsets. We study the structure and algebra of multirelations under the operations of union, intersection, sequential and parallel composition, as…
A subset $U$ of a set $S$ with a binary operation is called {\it avoidable} if $S$ can be partitioned into two subsets $A$ and $B$ such that no element of $U$ can be written as a product of two distinct elements of $A$ or as the product of…
We study the problem when every matrix over a division ring is representable as either the product of traceless matrices or the product of semi-traceless matrices, and also give some applications of such decompositions. Specifically, we…
Motivated by recent appearance of multivalued structures in categorification, tropical geometry and other areas, we study basic properties of abstract multisemigroups. We give many new and old examples and general constructions for…
A semiring can be ``completed'' (i.e., embedded into a semiring in which all infinite sums are defined and satisfy some reasonable properties) iff this semiring can be naturally partially ordered. This construction is ``natural'' (a left…
A ringoid is a set with two binary operations that are linked by the distributive laws. We study special classes of ringoids that are congruence-simple or ideal-simple. In particular, we examine generalised parasemifields and…
Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an…
We define the concept of a bi-operad. We develop the homotopy theory of "Bital-Sets" and of infinite-bi-operads. We develop a geometry of generalized schemes based on the spectra of distributive monochromatic bi-operads.
Diagram semigroups are interesting algebraic and combinatorial objects, several types of them originating from questions in computer science and in physics. Here we describe diagram semigroups in a general framework and extend our…
Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a…
A \emph{loop} $(B,\cdot)$ is a set $B$ together with a binary operation $\cdot$ such that (i) for each $a\in B$, the left and right translation mappings $L_{a}:B\to B: x \mapsto a\cdot x$ and $R_{a}:B\to B: x \mapsto x\cdot a$ are…