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A survey of some open problems on \(p_ n\)-sequences and free spectra of algebras and varieties. (English) Zbl 0772.08001

Universal algebra and quasigroup theory, Lect. Conf., Jadwisin/Pol. 1989, Res. Expo. Math. 19, 57-88 (1992).
[For the entire collection see Zbl 0745.00017.]
For an algebra \({\mathbf A}\), let \(f_ n({\mathbf A})\) denote the size of the free algebra on \(n\) generators over \({\mathbf A}\). The free spectrum of \({\mathbf A}\) is the sequence \[ f({\mathbf A})=\bigl\langle f_ 0({\mathbf A}),f_ 1({\mathbf A}),\dots,f_ n({\mathbf A}),\dots\bigl\rangle. \] Let \(p_ n({\mathbf A})\) denote the number of essentially \(n\)-ary operations composed from the \(n\)-ary projection operations using the basic operations of \({\mathbf A}\) and which depend on all \(n\) variables. The \(p_ n\)-sequence of \({\mathbf A}\) is the sequence \[ p({\mathbf A})=\bigl\langle p_ 0({\mathbf A}),p_ 1({\mathbf A}),\dots,p_ n({\mathbf A}),\dots\bigr\rangle. \] The free spectrum and the \(p_ n\)-sequence reflect the same properties of \({\mathbf A}\). A simple combinatorial formula expresses the \(p_ n\)-sequence from the free spectrum, and vice versa.
The study of \(p_ n\)-sequences originates with the papers of W. Sierpiński in the fourties. The \(p_ n\)-sequence was defined by E. Marczewski in the sixties. A great deal of work was done in the sixties by Marczewski and his coworkers in Wroclaw. In the late sixties this work was continued in Manitoba. A survey article was published in 1970 by the first author [Proc. Conf. Universal Algebra, Queen’s Univ., Kingston, 1969, 1-106 (1970; Zbl 0235.08003)].
In the two decades since this survey, substantial progress has been made. This article attempts to survey the major results and list the most interesting open problems. Of the many results surveyed, there are 31 formally stated as theorems. There are 29 open problems. The survey concludes with a list of 136 references.

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
08B20 Free algebras
08-02 Research exposition (monographs, survey articles) pertaining to general algebraic systems
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