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A350876
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a(n) is the number of nonisomorphic flexible left-right-alternative magmas with n elements. That is, a(n) is the number of nonisomorphic magmas with n elements which satisfy all of the identities x(yx) = (xy)x, (xx)y = x(xy), and x(yy) = (xy)y (for all x and y).
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2
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OFFSET
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0,3
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COMMENTS
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Compare A350873 and A350875, which are the numbers of flexible magmas with n elements and left-right-alternative magmas with n elements (up to isomorphism). The fact that a(n) < A350875(n) for n >= 3 means that left-right-alternativity for magmas (the identities (xx)y = x(xy) and x(yy) = (xy)y) does not imply flexibility (x(yx) = (xy)x). This is in contrast to the situation for non-associative rings, where this implication does hold (due to the additional additive structure).
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LINKS
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EXAMPLE
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There are 10 nonisomorphic magmas with 2 elements, 5 of which are flexible and left-right-alternative, so a(2) = 5.
Similarly there are 3330 nonisomorphic magmas with 3 elements, 33 of which satisfy all of (xy)x = x(yx), (xx)y = x(xy), and x(yy) = (xy)y for all x and y, so a(3) = 33.
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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