Variants and satisfiability in the infinitary unification wonderland

So far, results about variants, the finite variant property (FVP), variant unification, and variant satisfiability have been developed for equational theories $E\cup B$ where B is a set of axioms having a finitary unification algorithm, and the equations E, oriented as rewrite rules $\overrightarrow{E}$, are convergent modulo B. The extension to the case when B has an infinitary unification algorithm, for example because of non-commutative symbols having associative axioms, was not developed. This paper develops such an extension. In particular, the relationships between the FVP and the boundedness (BP) properties, the identification of conditions on $E\cup B$ ensuring FVP, the effective computation of variants and variant unifiers, and criteria making possible the existence of variant satisfiability procedures for the initial algebras of theories $E\cup B$ that are either FVP or BP are all explored in detail. The extension from the finitary to the infinitary B-unification case includes some surprises. Furthermore, since all the results are extended beyond FVP theories to the wider class of BP theories, new opportunities are opened up to use these symbolic techniques in wider classes of theories and applications.