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On the Zappa-Szép product. (English) Zbl 1078.20062
The well-known direct product of two groups requires that both factors are normal in the product. This was generalized to the semidirect product in which one only of the factors is required to be normal. In 1942, Zappa introduced and developed the Zappa-Szép product in which neither factor is required to be normal. Many variations of the product were given and studied by Rédei and others and the products were used to discover properties of groups. Similar products in other settings were studied by Szép (1958, 1962). Through his work on a family of closely related groups, known as Thompson’s groups (whose structure as groups of fractions of monoids were mostly Zappa-Szép products of simpler monoids) led him to the fact that there is an intimate relation between Thompson’s groups and certain categories. This relationship is helped by the fact that the Zappa-Szép product works well with categories.
The author discovers that group-like properties such as associativity, fullness of multiplication, etc. can be assumed or removed and the Zappa-Szép product can still be used at some level. The paper is a record of the author’s observations on the Zappa-Szép properties that are needed to work with categories, monoids and groups of fractions of monoids.

##### MSC:
 20N02 Sets with a single binary operation (groupoids) 20E22 Extensions, wreath products, and other compositions of groups 20M10 General structure theory for semigroups 20M50 Connections of semigroups with homological algebra and category theory 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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