Shirshov, A. I. Subalgebras of free commutative and free anticommutative algebras. (Die Unteralgebren der freien kommutativen und der freien antikommutativen Algebren.) (Russian) Zbl 0055.02703 Mat. Sb., N. Ser. 34(76), 81-88 (1954). For brevity refer to commutative and anticommutative algebras as \(K\)-algebras and \(AK\)-algebras, respectively. Then the author proves the proposition: “Every subalgebra of a free \(\varepsilon\)-algebra is again a free \(\varepsilon\)-algebra” for \(\varepsilon = K, AK\). The method of proof follows closely that of the author’s paper [Mat. Sb., N. Ser. 33(75), 441–452 (1953)] reviewed in Zbl 0052.03004. If \(R\) is any set, the \(\varepsilon\)-regular words in \(R\) are defined analogously to the regular words in a free Lie algebra (loc. cit.): The \(\varepsilon\)-regular words of length 1 are the nonassociative words of length 1 (i. e. the elements of \(R\) itself) ordered in any way.If the \(\varepsilon\)-regular words of length \(< n\) have been defined and ordered (so that all words of length \(< m\) precede all words of length \(m\)), then a word \(w\) of length \(n\) is called \(\varepsilon\)-regular, if 1. \(w = u v\), where \(u\) and \(v\) are \(\varepsilon\)-regular, and 2. \(u \geq v\) for \(\varepsilon = K\); \(u > v\) for \(\varepsilon = AK\).The \(\varepsilon\)-regular words of length \(n\) are then ordered in any way to follow the \(\varepsilon\)-regular words of length \(< n\).Let \(\mathfrak A\) be the free \(\varepsilon\)-algebra on the set \(R\) over a field \(F\) (of characteristic \(\neq 2\) if \(\varepsilon= AK\)). Then the \(\varepsilon\)-regular words form a basis of \(\mathfrak A\).For any subalgebra \(\mathfrak B\) of \(\mathfrak A\) ( a set \(\mathfrak M\) can be defined in exactly the same way as for a Lie algebra (loc. cit.) and this is shown to be a set of free generators of \(\mathfrak B\); hence \(\mathfrak B\) is a free \(\varepsilon\)-algebra. Reviewer: Paul M. Cohn (Manchester) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 3 ReviewsCited in 40 Documents MSC: 17A50 Free nonassociative algebras Citations:Zbl 0052.03004 PDFBibTeX XMLCite \textit{A. I. Shirshov}, Mat. Sb., Nov. Ser. 34(76), 81--88 (1954; Zbl 0055.02703) Full Text: EuDML