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Abstract.In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform solution to the word problems for free objects in all varieties of commutative semigroups.

Abstract. A clone C on a set A is a set of operations on A containing the projection operations and closed under composition. A combinatorial invariant of a clone is its /^-sequence {pn{C), px(C), ...), where pn{C) is the number of essentially n-ary operations in C. We investigate the links between this invariant and structural properties of clones. It has been conjectured that the pn -sequence of a clone on a finite set is either eventually strictly increasing or is bounded above by a finite constant. We verify this conjecture for a large family of clones. A special role in our work is played by totally symmetric operations and totally symmetric clones. We show that every totally symmetric clone on a finite set has a bounded pn -sequence and that it is decidable if a clone is totally symmetric.

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