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Sets with two associative operations. (English) Zbl 1032.16032
The author considers sets equipped with two associative binary operations, calling such a structure a duplex. Relevant examples of duplexes are dimonoids, defined by J.-L. Loday [C. R. Acad. Sci., Paris, Sér. I 321, No. 2, 141-146 (1995; Zbl 0845.16036)]. An example of a duplex which is not a dimonoid is given by using permutations. A free duplex generated by a given set is constructed via planar trees. Duplexes coming from planar binary trees and vertices of cubes are discussed.

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17A32 Leibniz algebras
20M05 Free semigroups, generators and relations, word problems
08B20 Free algebras
05C05 Trees
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