This article is cited in 6 papers

RESEARCH ARTICLE

Structure of relatively free trioids

A. V. Zhuchok

Department of Algebra and System Analysis, Luhansk Taras Shevchenko National University, Gogol Square, 1, Starobilsk 92703, Ukraine

Abstract: Loday and Ronco 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.

Keywords: trioid, trialgebra, free trioid, free trialgebra, relatively free trioid, semigroup.

MSC: 08B20, 20M10, 20M50, 17A30, 17D99

Received: 30.11.2020

Language: English

DOI: 10.12958/adm1732