Small systems that satisfy the Average axioms. AA = A AB = BA AB.CD = AC.BD elements systems tables group orders 1 1 1 1 2 1 2 1 3 3 10 1 2 6 4 7 92 1.2 2.3 6.2 5 22 1321 1.5 2.10 4 6.4 20 24 6 77 27882 1.19 2.31 4.7 6.12 12.4 20 24.2 120 7 314 819330 1.85 2.122 4.32 6.36 8.4 12.19 20.2 24.6 36.2 42 48 72 120.2 720 Elements is the size of the system. Systems is the count of non-isomorphic systems of that size. Tables is the total number of tables, with no culling for isomorphism. Group orders is the number of systems with each size of automorphism group. For example, there are 314 non-isomorphic Average systems with 7 elements. 85 of those systems have the trivial automorphism group (only the identity), and each system gives rise to 7! = 5040 distinct tables. There's one system with an automorphism group of 720 elements, which gives rise to only 5040/720 = 7 different tables. The total number of possible 7-element tables is 7^49, of which roughly 7^7 satisfy the Average rules. We have the obvious identities 314 = 85 + 122 + 32 + ... + 1 + 2 + 1, and 819330 = 5040 * (85/1 + 122/2 + 32/4 + ... + 1/72 + 2/120 + 1/720). "Minimum" satisfies the rules, both for integers and in a lattice. Also, any idempotent commutative semigroup is also Average. (Idempotent commutative semigroups are the same as partial orders with well-defined GLB operations.) It looks like, for odd primes P, the number of tables is 1 mod P. The number of tables seems to be around N^N. The number of systems seems to be around (N+1)!/2^N. Rich rcs@cs.arizona.edu