arXiv subjects

Anna B. Romanowska

Publications and source records attributed to Anna B. Romanowska.

5 recordsLinked to original sources

arXiv2026

Semilattice sums of algebras and Mal'tsev products of varieties

The Mal'tsev product of two varieties of similar algebras is always a quasivariety. We consider the question of when this quasivariety is a variety. The main result asserts that if $\mathcal{V}$ is a strongly irregular variety with no nullary operations and at least one non-unary operation, and $\mathcal{S}$ is the variety, of the same type as $\mathcal{V}$, equivalent to the variety of semilattices, then the Mal'tsev product $\mathcal{V} \circ \mathcal{S}$ is a variety. It consists precisely of semilattice sums of algebras in $\mathcal{V}$. We derive an equational base for the product from an equational base for $\mathcal{V}$. However, if $\mathcal{V}$ is a regular variety, then the Mal'tsev product may not be a variety. We discuss various applications of the main result, and examine some detailed representations of algebras in $\mathcal{V} \circ \mathcal{S}$.

math.RA↗

arXiv2026

On the intrinsic geometry of polyhedra: Convex polygon coordinates

There is a very extensive literature dealing with convex polytopes from the standpoints of combinatorics and numerical analysis. By contrast, the current paper adopts an alternative viewpoint that regards a polytope as an autonomous space in its own right, with its own intrinsic geometry. Our attention is focused on the complete set of all the coordinate systems that serve to locate a point of the polytope; divorced, for example, from the smoothness issues that are of concern for applications in numerical analysis. We use the efficient and appropriate algebraic language of barycentric algebras to elicit the convex structure of the set of polytope coordinate systems. Specializing to convex polygons, we examine the chordal coordinate systems that are determined by the triangulations of the polygon. An algorithm to compute the coordinates of a point within such a system is presented. The algorithm relies on a coalgebra structure that transports a probability distribution on one side of a triangle to distributions on each of the remaining two sides. The Catalan number enumeration of the polygon triangulations (well-known within combinatorics) is then obtained in a natural geometric fashion from the parsing trees of the coalgebra structure of our algorithm.

math.MG↗

arXiv2025

From affine to barycentric coordinates in polytopes

Each point of a simplex is expressed as a unique convex combination of the vertices. The coefficients in the combination are the barycentric coordinates of the point. For each point in a general convex polytope, there may be multiple representations, so its barycentric coordinates are not necessarily unique. There are various schemes to fix particular barycentric coordinates: Gibbs, Wachspress, cartographic, etc. In this paper, a method for producing sparse barycentric coordinates in polytopes will be discussed. It uses a purely algebraic treatment of affine spaces and convex sets, with barycentric algebras. The method is based on a certain decomposition of each finite-dimensional convex polytope into a union of simplices of the same dimension.

math.MG↗

arXiv2024

Mal'tsev products of varieties, I

We investigate the Mal'tsev product $\mathcal{V} \circ \mathcal{W}$ of two varieties $\mathcal{V}$ and $\mathcal{W}$ of the same similarity type. Such a product is usually a quasivariety but not necessarily a variety. We give an equational base for the variety generated by $\mathcal{V} \circ \mathcal{W}$ in terms of identities satisfied in $\mathcal{V}$ and $\mathcal{W}$. Then the main result provides a new sufficient condition for $\mathcal{V} \circ \mathcal{W}$ to be a variety: If $\mathcal{W}$ is an idempotent variety and there are terms $f(x,y)$ and $g(x,y)$ such that $\mathcal{W}$ satisfies the identity $f(x,y) = g(x,y)$ and $\mathcal{V}$ satisfies the identities $f(x,y) = x$ and $g(x,y) = y$, then $\mathcal{V} \circ \mathcal{W}$ is a variety. We also provide a number of examples and applications of this result.

math.RA↗

arXiv2024

Mal'tsev products of varieties, II

The Mal'tsev product of two varieties of the same similarity type is not in general a variety, because it can fail to be closed under homomorphic images. In the previous paper we provided a new sufficient condition for such a product to be a variety. In this paper we extend that result by weakening the assumptions regarding the two varieties. We also explore the various special cases of our new result and provide a number of examples of its application.

math.RA↗