arXiv subjects

Patrick Wynne

Publications and source records attributed to Patrick Wynne.

2 recordsLinked to original sources

arXiv2026

Structure and Complexity of 2-Nilpotent Mal'cev Algebras

We investigate the structure of central extensions for algebras in a congruence modular variety. We use a multisorted algebraic object called a clonoid to understand the term clone of such a central extension. We develop the difference clonoid of such a central extension and use it to show that the number of $2$-step nilpotent algebras on a fixed finite set is finite if and only if the set is of squarefree order. The subpower membership problem for a finite algebraic structure $\mathbb{A}$ is the problem of deciding on input $a_1,\dots,a_k, b \in A^n$, whether $b$ is in the subalgebra of $\mathbb{A}^n$ generated by $a_1, \dots, a_k$. We show that for a large class of nilpotent Mal'cev algebras the subpower membership problem is solvable in polynomial time, in particular for $2$-step nilpotent Mal'cev algebras of squarefree order.

math.RA↗

arXiv2023

Clonoids between modules

Clonoids are sets of finitary functions from an algebra $\mathbb{A}$ to an algebra $\mathbb{B}$ that are closed under composition with term functions of $\mathbb{A}$ on the domain side and with term functions of $\mathbb{B}$ on the codomain side. For $\mathbb{A},\mathbb{B}$ (polynomially equivalent to) finite modules we show: If $\mathbb{A},\mathbb{B}$ have coprime order and the congruence lattice of $\mathbb{A}$ is distributive, then there are only finitely many clonoids from $\mathbb{A}$ to $\mathbb{B}$. This is proved by establishing for every natural number $k$ a particular linear equation that all $k$-ary functions from $\mathbb{A}$ to $\mathbb{B}$ satisfy. Else if $\mathbb{A},\mathbb{B}$ do not have coprime order, then there exist infinite ascending chains of clonoids from $\mathbb{A}$ to $\mathbb{B}$ ordered by inclusion. Consequently any extension of $\mathbb{A}$ by $\mathbb{B}$ has countably infinitely many $2$-nilpotent expansions up to term equivalence.

math.RA↗