Partial orders and the axiomatic theory of shuffle (pomsets)

The shuffle operator has been fairly extensively studied as a model of concurrency in the context of regular expressions. However, very little study has been done on the algebraic properties of shuffle and its analogue to Kleene closure, which we call shuffle closure. In this thesis we address the issue of finite axiomatizability of various combinations of concatenation, shuffle, and addition operators under two different models: the traditional model of formal languages and the alternate model of partially-ordered multisets. Partially-ordered multisets, or pomsets, are a generalization of strings that allows the symbols to be partially ordered instead of totally ordered as is the case in a string. Included in this thesis is a study of the structure of pomsets and pomset-definable operations. One concept that arises is that of subsumption, which occurs when one pomset is a generalization of another. The idea of subsumption leads to a definition of pomset ideals, which are the pomset analogues to languages. The study of pomsets also leads us to explore the dualism between shuffle and concatenation. We show, among other results, that the theory of sets of pomsets under addition, shuffle, and concatenation is finitely axiomatizable. In addition, the theory of sets of pomsets under any operations including Kleene closure or shuffle closure is not finitely axiomatizable, generalizing a proof of Conway's. Another important result is that languages of strings of length two are sufficient to distinguish pomsets which are not language-equivalent.