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%I #6 Aug 05 2019 07:37:12
%S 0,1,2,4,5,6,8,16,17,24,32,34,40,64,65,66,68,69,70,72,80,81,85,88,96,
%T 98,102,104,120,128,256,257,384,512,514,640,1024,1025,1026,1028,1029,
%U 1030,1152,1280,1281,1285,1408,1536,1538,1542,1664,1920,2048,2056,2176
%N BII-numbers of set-systems (without {}) closed under union and intersection.
%C A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.
%C A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
%e The sequence of all set-systems closed under union and intersection together with their BII-numbers begins:
%e 0: {}
%e 1: {{1}}
%e 2: {{2}}
%e 4: {{1,2}}
%e 5: {{1},{1,2}}
%e 6: {{2},{1,2}}
%e 8: {{3}}
%e 16: {{1,3}}
%e 17: {{1},{1,3}}
%e 24: {{3},{1,3}}
%e 32: {{2,3}}
%e 34: {{2},{2,3}}
%e 40: {{3},{2,3}}
%e 64: {{1,2,3}}
%e 65: {{1},{1,2,3}}
%e 66: {{2},{1,2,3}}
%e 68: {{1,2},{1,2,3}}
%e 69: {{1},{1,2},{1,2,3}}
%e 70: {{2},{1,2},{1,2,3}}
%e 72: {{3},{1,2,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union@@@Tuples[bpe/@bpe[#],2]]&&SubsetQ[bpe/@bpe[#],Intersection@@@Tuples[bpe/@bpe[#],2]]&]
%Y Cf. A048793, A102894, A102895, A102896, A102897, A326031, A326875, A326876, A326878, A326880, A326901.
%K nonn
%O 1,3
%A _Gus Wiseman_, Aug 04 2019
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