## Rocky Mountain Journal of Mathematics

### Semigroup compactifications of Zappa products

#### Abstract

A group $G$ with subgroups $S$ and $T$ satisfying $G = ST$ and $S\cap T = \{e\}$ gives rise to functions $[t,s] \in S$ and $\\lt t,s\> \in T$ such that $(st)(s't') = (s[t,s'])(\\lt t,s'\>t')$. This notion may be extended to arbitrary semigroups $S$, $T$ with identities, producing the Zappa product of $S$ and $T$, a generalization of direct and semidirect product. Necessary and sufficient conditions are given for a semigroup compactification of a Zappa product $G$ of topological semigroups $S$ and $T$ to be canonically isomorphic to a Zappa product of compactifications of $S$ and $T$. The result is applied to various types of compactifications of $G$, including the weakly almost periodic and almost periodic compactifications.

#### Article information

Source
Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1903-1921.

Dates
First available in Project Euclid: 2 February 2015

https://projecteuclid.org/euclid.rmjm/1422885100

Digital Object Identifier
doi:10.1216/RMJ-2014-44-6-1903

Mathematical Reviews number (MathSciNet)
MR3310954

Zentralblatt MATH identifier
1311.22003

#### Citation

Junghenn, H.D.; Milnes, P. Semigroup compactifications of Zappa products. Rocky Mountain J. Math. 44 (2014), no. 6, 1903--1921. doi:10.1216/RMJ-2014-44-6-1903. https://projecteuclid.org/euclid.rmjm/1422885100

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