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The lattice of equational classes of commutative semigroups does not satisfy any special lattice laws. In [4] R. Schwabauer proved that the lattice £ of equational classes of commutative semigroups is nonmodular. In this paper we will prove a maximal extension of this result, namely, £ does not satisfy any special lattice laws. The free commutative semigroup on countably many generators, 5(0)), is the set of sequences (un)neN oí nonnegative integers, such that m„ = 0 for all but finitely many nEN and £ un^l, with component-wise addition. [For convenience, we write (a,) for (un)neN-] A commutative semigroup equation is a pair Hun), ivn)) of elements of $(«) (see [l]). A commutative semigroup (S, •) satisfies the equation ((un), (vn)) iff, for every family (an) of elements of S, n{oî-|«»3^0} = U{an'\vn^0}. A set 2 of equations is closed [l, p. 170, Definition 2] iff it contains all trivial equations, is symmetric and transitive, and is closed under multiplication and substitution of terms for variables. Thus 2 is closed iff it satisfies the following conditions : (PI): ((«»),(«,)) G2 for all («»)€$(«)• (P2) : If ((«.), (O) ES then ((».), («„)) G2. (P3): If ((«,), (v„))GS and ((v„), (0)G2 then ((«„), (0)G2. (P4): If ((«.), (O)GS and («), «))G2 then ((«,+<), (»„+<)) G2. (P5): If ((«„), K))GS, (kn)E%(w) and pEN, then the result of "substituting (kH) for the pth variable" in ((un), (vn)) is in 2, i.e., ((wn), (x„))G2 where wn = Un + knup for n^p, wp = kpup, and xn=vn 4knvp for U9±p and xp = kpVp. (Note that these conditions (PI) to (P5) are, essentially, a restatement of conditions (i) to (v) in Grätzer [l, p. 170, Definition 2]; in condition (iv) we need only consider the one binary operation, hence Received by the editors May 14, 1970. AMS 1969 subject classifications. Primary 2090, 0630; Secondary 0830.

Embedding the dual of 𝜋_{𝑚} in the lattice of equational classes of commutative semigroups