In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ) of automorphisms of the category Θ of finitely generated free algebras of Θ is of great importance. In this paper, we prove that all automorphisms of categories of free S-acts are semi-inner, which solves a variation of Problem 12 in [12] for monoids. We also give a description of automorphisms of categories of finit...

We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.

A Rota–Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota–Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota–Baxter algebras have been for commutative algebras. Free commutative Rota–Baxter algebras were constructed by Rota and Cartier in the 1970s. A later construction was obta...

We interoduce a new characterization of algebras of normal forms of term rewriting systems [35] as algerbras of term free in itself (any function from free generators into the algebra generates endomorphism of the algebra). Introduced algebras are free in classes of algebras satisfying some sets of equalities. Their universes are subsets of all terms and the denotations of operation symbols are...

Completely iterative algebras (cias) are those algebras in which recursive equations have unique solutions. In this paper we study complete iterativity for algebras with computational effects (described by a monad). First, we prove that for every analytic endofunctor on Set there exists a canonical distributive law over any commutative monad M , hence a lifting of that endofunctor to the Kleisl...