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Structure of relatively free trioids. (English) Zbl 1491.08013

Summary: J.-L. Loday and M. Ronco [Contemp. Math. 346, 369–398 (2004; Zbl 1065.18007)] introduced the notions of a trioid and a trialgebra, and constructed the free trioid of rank \(1\) and the free trialgebra. This paper is a survey of recent developments in the study of free objects in the varieties of trioids and trialgebras. We present the constructions of the free trialgebra and the free trioid, the free commutative trioid, the free \(n\)-nilpotent trioid, the free left (right) \(n\)-trinilpotent trioid, and the free rectangular trioid. Some of these results can be applied to constructing relatively free trialgebras.

MSC:

08B20 Free algebras
08A62 Finitary algebras
20M10 General structure theory for semigroups
17A30 Nonassociative algebras satisfying other identities
17A50 Free nonassociative algebras

Citations:

Zbl 1065.18007
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References:

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