Finite GE-Algebras of Order at Most 6

Source abstract

Up to isomorphism there are 1, 3, 21, 479 and 40,791 GE-algebras of orders 2 to 6. Two separately written programs give the counts, checked with an SMT solver up to order 5. We give the counts for six named subclasses (transitive, commutative, left exchangeable, belligerent, antisymmetric and Hilbert; two of the columns were known) and, for every non-inclusion between them, the least order of a separating example together with the number of minimal examples. The nine patterns of properties that occur are exactly those allowed by the published inclusions, which therefore axiomatize the Boolean theory of the six properties. Finally, a GE-algebra fails the exchange identity exactly when two of its rows are different constant maps on a two-element set; hence every GE-algebra with at most four elements is left exchangeable, and the bound is sharp.