The inclusion structure of partially lossy queue monoids and their trace submonoids

We model the behavior of a lossy fifo-queue as a monoid of transformations that are induced by sequences of writing and reading. To have a common model for reliable and lossy queues, we split the alphabet of the queue into two parts: the forgettable letters and the letters that are transmitted reliably. We describe this monoid by means of a confluent and terminating semi-Thue system and then study some of the monoid’s algebraic properties. In particular, we characterize completely when one such monoid can be embedded into another as well as which trace monoids occur as submonoids. Surprisingly, these are precisely those trace monoids that embed into the direct product of two free monoids – which gives a partial answer to a question raised by Diekert et al. at STACS 1995.

Non-generable formal languages

A set A of words on a finite alphabet Σ is said to be generable if it is the closure of a computable inductive operator; in particular, if S is a semi-Thue system then the set of words derivable in S is generable. An RE set of words (equivalently, a phrase-structure or type 0 language in the sense of Chomsky [Information and Control 2 (1959), 137–167]) which is non-generable is constructed by means of a finite injury priority argument. The construction is refined to obtain a non-generable set of degree 0′ and, for any degree d, a non-generable set of degree ⩽ d.