arXiv subjects
Publications and source records attributed to Bill de Witt.
We show that the monoids totM_{k,1} introduced by Birget and their generalizations tot nM_{k,r} which extend the Brin-Higman-Thompson groups, can be realized as the endomorphism monoids of higher-dimensional Jónsson-Tarski algebras. We also show how elements of these monoids can be thought of as "rewrite rules". We use these representations to show that the monoids are finitely presented.
A finite unary algebra $(A,F)$ has only countably many countable subdirect powers if and only if every operation $f\in F$ is either a permutation or a constant mapping.
In this paper we discuss the relationship between direct products of monounary algebras and their components, with respect to the properties of residual finiteness, strong/weak subalgebra separability, and complete separability. For each of these properties $\mathcal{P}$, we give a graphical criterion $\mathcal{C_P}$ such that a monounary algebra $A$ has property $\mathcal{P}$ if and only if it satisfies $\mathcal{C_P}$. We also show that for a direct product $A\times B$ of monounary algebras, $A\times B$ has property $\mathcal{P}$ if and only if one of the following is true: either both $A$ and $B$ have property $\mathcal{P}$, or at least one of $A$ or $B$ are backwards-bounded, a special property which dominates direct products and which guarantees all $\mathcal{P}$ hold.