Free Idempotent Monoid

A free idempotent monoid is a monoid that satisfies the identity x^2=x and is generated by a set of elements. If the generating set of such a monoid is finite, then so is the free idempotent monoid itself. The number of elements in the monoid depends on the size of the generating set, and the size the generating set uniquely determines a free idempotent monoid. On zero letters, the free idempotent monoid has one element (the identity). With one letter, the free idempotent monoid has two elements (1,a). With two letters, it has seven elements: (1,a,b,ab,ba,aba,bab). In general, the numbers of elements in the free idempotent monoids on n letters are 1, 2, 7, 160, 332381, ... (OEIS A005345). These are given by the analytic expression

 sum_(k=0)^n(n; k)product_(i=1)^k(k-i+1)^(2^i),

where (k; n) is a binomial coefficient. The product can be done analytically, giving the sum

 sum_(k=0)^n(n; k)exp{2[2^kd/(dn)Li_n(2)|_(n=0)-d/(ds)Phi(2,s,-k)|_(s=0)]}

in terms of derivatives of the polylogarithm Li_n(2) with respect to its index and the Lerch transcendent Phi(2,s,-k) with respect to its second argument.

See also


Portions of this entry contributed by Todd Rowland

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Berstel, J. and Reutenauer, C. In Combinatorics on Words (Ed. M. Lothaire). Cambridge, England: Cambridge University Press, p. 32, 1997.Green, J. and Rees, D. "On Semigroups in which x^r=x." Math. Proc. Cambridge Philos. Soc. 48, 35-40, 1952.Lallement, G. Semigroups and Combinatorial Applications. New York: Wiley, 1979.

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Free Idempotent Monoid

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Free Idempotent Monoid." From MathWorld--A Wolfram Web Resource.

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