Nominal Monoids

Abstract

We develop an algebraic theory for languages of data words. We prove that, under certain conditions, a language of data words is definable in first-order logic if and only if its syntactic monoid is aperiodic.

Recognisable Languages over Monads

“…This setting was studied in [Boj13], although not using the monad terminology. The Syntactic Morphism Theorem holds in this setting, as was shown in Lemmas 3.3 and 3.4 of [Boj13]. We will show that the Pseudovariety Theorem fails in this setting.…”

“…To complete the definition of the setting, define finite alphabets to be finitely supported sets which are orbit-finite, and define finite algebras to be finitely supported orbit-finite semigroups. This setting was studied in [Boj13], although not using the monad terminology. The Syntactic Morphism Theorem holds in this setting, as was shown in Lemmas 3.3 and 3.4 of [Boj13].…”

“…The following lemma is in the category of sets, or more generally, in categories which have a powerset functor that preserves finiteness. A non-example is the category of nominal sets with orbit-finite sets, where powerset does not preserve orbit-finiteness, and also mso contains non-recognisable languages, see [Boj13]. Lemma 6.2 If L contains only T-recognisable T-languages, then so does mso T (L).…”

Equational Axiomatization of Algebras with Structure

“…In the quantitative setting, equations s = ε t are equipped with a non-negative real number ε, interpreted as "s and t have distance at most ε". (3) Nominal algebras (given by a nominal set and equivariant operations) are used in the theory of name binding [24] and have proven useful for characterizing logics for data languages [9,11]. Varieties of nominal algebras were studied by Gabbay [13] and Kurz and Petrişan [16].…”