@inbook {IOPORT.05776430,
    author       = {Kl{\'\i}ma, Ond\v{r}ej and Pol\'ak, Libor},
    title        = {On Sch\"utzenberger products of semirings.},
    year         = {2010},
    booktitle    = {Developments in language theory. 14th international conference, DLT 2010, London, ON, Canada, August 17--20, 2010. Proceedings},
    isbn         = {978-3-642-14454-7},
    pages        = {279-290},
    publisher    = {Berlin: Springer},
    doi          = {10.1007/978-3-642-14455-4_26},
    abstract     = {In some earlier papers, the authors studied the recognition of languages by finite idempotent semirings and so-called disjunctive varieties of languages that stand in an Eilenberg-type correspondence to the pseudovarieties of finite idempotent semirings. Here they introduce a Sch\"utzenberger product of such semirings. Let $S_0,S_1,\dots, S_n$ be finite idempotent semirings and let $S$ be their Sch\"utzenberger product. It is shown that { indent6.5mm \item{(1)} $S$ is a finite idempotent semiring; \item{(2)} if $L_i$ is a language recognized by $S_i$ $(i= 0,1,\dots,n)$ and $a_1,\dots,a_n$ are letters, then $L_0a_1L_1a_2\cdots a_nL_n$ is recognized by $S$; \item{(3)} any language recognized by $S$ is a `polynomial' of the languages recognized by the original semirings, i.e., the union of finitely many languages $L_{i_0}a_1L_{i_1}a_2\cdots a_kL_{i_k}$, where $0\le i_0<\dots< i_k\le n$, $a_1,\dots, a_k$ are letters, and $L_{i_m}$ is a language recognized by $S_{i_m}$ $(m= 0,\dots,k)$. } As applications of these results, the authors consider the pseudovarieties of finite semirings that correspond to disjunctive varieties of certain simple languages of polynomial form, and they show that in each case the pseudovariety is generated by a single Sch\"utzenberger product constructed from the trivial semiring or from the two-element idempotent semiring. Finally, they show that the polynomial closure of any disjunctive variety $\cal V$ is a disjunctive variety, and that the corresponding pseudovariety is generated by Sch\"utzenberger products of semirings belonging to the pseudovariety that corresponds to ${\Cal V}$.},
    reviewer     = {Magnus Steinby (Turku)},
    identifier   = {05776430},
}