International Journal of Algebra and Computation

We investigate notions of complete representation by partial functions, where the operations in the signature include antidomain restriction and may include composition, intersection, update, preferential union, domain, antidomain, and set difference. When the signature includes both antidomain restriction and intersection, the join-complete and the meet-complete representations coincide. Otherwise, for the signatures we consider, meet-complete is strictly stronger than join-complete. A necessary condition to be meet-completely representable is that the atoms are separating. For the signatures we consider, this condition is sufficient if and only if composition is not in the signature. For each of the signatures we consider, the class of (meet-)completely representable algebras is not axiomatizable by any existential-universal-existential first-order theory. For 14 expressively distinct signatures, we show, by giving an explicit representation, that the (meet-)completely representable algebras form a basic elementary class, axiomatizable by a universal-existential-universal first-order sentence. The signatures we axiomatize are those containing antidomain restriction and any of intersection, update, and preferential union and also those containing antidomain restriction, composition, and intersection and any of update, preferential union, domain, and antidomain.

If A is an associative algebra, then we can define the adjoint Lie algebra $A^{(-)}$ and Jordan algebra $A^{(+)}$. It is easy to see that any associative Rota–Baxter (RB) operator on A induces a Lie and Jordan RB operator on $A^{(-)}$ and $A^{(+)}$, respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra $H_4$, which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra $H_4^{(-)}$.

Rota–Baxter operators on groups have been recently defined in [L. Guo, H. Lang and Y.  Sheng, Integration and geometrization of Rota–Baxter Lie algebras, Adv. Math. 387 (2021) 107834], and they share a close connection with skew braces, as demonstrated in [V. Bardakov and V. Gubarev, Rota–Baxter groups, skew left braces, and the Yang–Baxter equation, J. Algebra 587 (2022) 328–351]. In this paper, we classify all Rota–Baxter operators of weight 1 on the Heisenberg Lie algebra of dimension 3 up to the Jordan canonical form of a $2\times 2$ matrix block by automorphisms of the Heisenberg Lie algebra. Using the fact that the exponential map from the Heisenberg Lie algebra to the Heisenberg group is bijective, we induce these operators on the Heisenberg group. Finally, we enumerate all skew left brace structures on the Heisenberg group induced by these Rota–Baxter operators.

We show that if a group contains ${\mathbb{Z}}^n\times F_m$ as a finite-index subgroup, then its cogrowth series is the diagonal of a rational function for every generating set. This answers a question of Pak and Soukup on the cogrowth of virtually abelian groups; and generalizes a result by Elder, Rechnitzer, Janse van Rensburg, and Wong on the cogrowth series of the Baumslag–Solitar groups $\mathrm{\text{BS}}(N,N)$.

We give a simple algorithm to solve the subgroup membership problem for virtually free groups given as a graph of finite groups. For a fixed virtually free group with a fixed generating set X, the subgroup membership problem is uniformly solvable in time $O(N{log}^{\ast }(N))$ where N is the sum of the word lengths of the inputs with respect to X. For practical purposes, this can be considered to be linear time. The algorithm is simple enough to work out small examples by hand and concrete examples are given to show how it can be used for computations in $\mathrm{\text{SL}}(2,\mathbb{Z})$ and $\mathrm{\text{GL}}(2,\mathbb{Z})$. We also give fast algorithm to decide whether a finitely generated subgroup of a virtually free group is free.

The algebra of ${\mathrm{\text{GL}}}_n$-invariants of m-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants are not known even for a pair of $2\times 2$ matrices. Working over an arbitrary field we classified all ${\mathrm{\text{GL}}}_2$-orbits on m-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\mathrm{\text{GL}}}_2$-invariant polynomial functions of m-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field.

In this paper, we investigate properties of varieties of algebras described by a novel concept of equation that we call commutator equation. A commutator equation is a relaxation of the standard term equality obtained substituting the equality relation with the commutator relation. Namely, an algebra $A$ satisfies the commutator equation $p{\approx }_{C}q$ if for each congruence $𝜃$ in $Con(A)$ and for each substitution $p^A,q^A$ of elements in the same $𝜃$-class, we have $(p^A,q^A)\in [𝜃,𝜃]$. This notion of equation draws inspiration from the definition of a weak difference term and allows for further generalization of it. Furthermore, we present an algorithm that establishes a connection between congruence equations valid in the variety generated by the abelian algebras of the idempotent reduct of a given variety and congruence equations that hold in the entire variety. Additionally, we provide a proof that if the variety generated by the abelian algebras of the idempotent reduct of a variety satisfies a nontrivial idempotent Mal’cev condition, then also the entire variety satisfies a nontrivial idempotent Mal’cev condition, a statement that follows also from [, Theorem 3.13].