Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Completeness Theorems for Pomset Languages and Concurrent Kleene
  Algebras   Pomsets constitute one of the most basic models of concurrency. A pomset is a
generalisation of a word over an alphabet in that letters may be partially
ordered. A term $t$ using the bi-Kleene operations $0,1, +, \cdot\, ,^*,
\parallel, ^{(*)}$ defines a language $ \mathopen{[\![ } t \mathclose{]\!] } $
of pomsets in a natural way.
  We prove that every valid universal equality over pomset languages using
these operations is a consequence of the equational theory of regular languages
(in which parallel multiplication and iteration are undefined) plus that of the
commutative-regular languages (in which sequential multiplication and iteration
are undefined). We also show that the class of $\textit{rational}$ pomset
languages (that is, those languages generated from singleton pomsets using the
bi-Kleene operations) is closed under all Boolean operations.
  An $ \textit{ideal}$ of a pomset $p$ is a pomset using the letters of $p$,
but having an ordering at least as strict as $p$. A bi-Kleene term $t$ thus
defines the set $ \textbf{Id} (\mathopen{[\![ } t \mathclose{]\!] }) $ of
ideals of pomsets in $ \mathopen{[\![ } t \mathclose{]\!] } $. We prove that if
$t$ does not contain commutative iteration $^{(*)}$ (in our terminology, $t$ is
bw-rational) then $\textbf{Id} (\mathopen{[\![ } t \mathclose{]\!] }) \cap
\textbf{Pom}_{sp}$, where $ \textbf{Pom}_{sp}$ is the set of pomsets generated
from singleton pomsets using sequential and parallel multiplication ($ \cdot$
and $ \parallel$) is defined by a bw-rational term, and if two such terms
$t,t'$ define the same ideal language, then $t'=t$ is provable from the Kleene
axioms for $0,1, +, \cdot\, ,^*$ plus the commutative idempotent semiring
axioms for $0,1, +, \parallel$ plus the exchange law $ (u \parallel v)\cdot ( x
\parallel y) \le v \cdot y \parallel u \cdot x $.

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