Dear Mathfun, Seqfans: Numbers of set systems of various types --------------------------------------- There are some sequences that may be missing from the OEIS, in case somebody would be interested in calculating them. This was suggested by reading pages 1 and 2 of the book by Heinz Koenig on Measure and Integration (Springer, 1997) We are considering a "set system" S, which is a nonempty collection of subsets A of a set X, where X has n points. There will be two versions of each sequence, depending on whether the points are labeled or unlabeled. We define several properties which S might have: U : A, B in S => A U B in S I : A, B in S => A intersect B in S \ : A, B in S with A contained in B => B\A in S - : A in S => complement of A in X is in S 0 : S contains the empty set 1 : S contains the universe set X We also define a new operation on subsets of X: |P|A|Q| = (P intersect complement of A) union (Q intersect A) [points in P but not in A plus points in both Q and A] Now Koenig defines 4 types of set systems: a lattice : properties U, I an oval : P,A,Q in S => |P|A|Q| in S a ring : properies U, I, \ an algebra : properties U, I, - The question is, how many lattices, ovals, rings, and algebras are there on n labeled/unlabeled points? (In fact one might consider any of the 2^7 combinations of the above 7 properies!) There are of course many sequences like these in the OEIS already, but usually there is no precise definition (this should be corrected!) so it is hard to tell which of these may be new. For example: posets: A000112 (unlabeled), A001035 (labeled) Lattices: A006966 (unlabeled); A055512 (labeled) Boolean lattices: A005493 Set systems of various types, studied recently by Mitch Harris and Don Knuth: A102894-A102897. E.g.: %I A102895 %S A102895 2,2,8,90,4542,2747402,151930948472 %N A102895 Number of ACI algebras or semilattices on n generators, with no identity element. %C A102895 An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation. %C A102895 Or, number of families of subsets of {1, ..., n} that are closed under intersection and contain the empty set. - this seems to count subsets with properties I, 0 in the labeled case %I A102897 %S A102897 2,4,14,122,4960,2771104,151947502948 %N A102897 Number of ACI algebras (or semilattices) on n generators. %C A102897 Also counts Horn functions on n variables, boolean functions whose set of truth assignments are closed under 'and', or equivalently, the boolean functions that can be written as a conjunction of Horn clauses, clauses with at most one negative literal. %C A102897 Also, number of families of subsets of {1,...,n} that are closed under intersection (because we can throw in the universe, or take it out, without affecting anything else). %C A102897 An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation. - this seems to count subsets with property I in the labeled case ? Likewise A102896 seems to count subsets with properties I, 1 in the labeled case and A102894 seems to count subsets with properties I, 0, 1 in the labeled case (see the full entries for more details) Neil