Dickson's Lemma, Higman's Theorem and Beyond: a survey of some basic results in order theory

We provide proofs for the fact that certain orders have no descending chains and no antichains.Comment: Survey pape

The Complexity of Checking Quasi-Identities over Finite Algebras with a Mal\u27cev Term

We consider finite algebraic structures and ask whether every solution of a given system of equations satisfies some other equation. This can be formulated as checking the validity of certain first order formulae called quasi-identities. Checking the validity of quasi-identities is closely linked to solving systems of equations. For systems of equations over finite algebras with finitely many fundamental operations, a complete P/NPC dichotomy is known, while the situation appears to be more complicated for single equations. The complexity of checking the validity of a quasi-identity lies between the complexity of term equivalence (checking whether two terms induce the same function) and the complexity of solving systems of polynomial equations. We prove that for each finite algebra with a Mal\u27cev term and finitely many fundamental operations, checking the validity of quasi-identities is coNP-complete if the algebra is not abelian, and in P when the algebra is abelian

Solving Systems of Equations in Supernilpotent Algebras

Recently, M. Kompatscher proved that for each finite supernilpotent algebra $\mathbf{A}$ in a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this algebra. Let $\mu$ be the maximal arity of the fundamental operations of $\mathbf{A}$, and let $$d := |A|^{\log_2 (\mu) + \log_2 (|A|) + 1}.$$ Applying a method that G. K\'{a}rolyi and C. Szab\'{o} had used to solve equations over finite nilpotent rings, we show that for $\mathbf{A}$, there is $c \in \mathbb{N}$ such that a solution of every system of $s$ equations in $n$ variables can be found by testing at most $c n^{sd}$ (instead of all $|A|^n$ possible) assignments to the variables. This also yields new information on some circuit satisfiability problems

Complexity of term representations of finitary functions

The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions, and we show that these bounds are often robust against the change of the basic operations of the structure