_Jonathan Vos Post (jvospost3(AT)gmail.com), _, Feb 13 2011

editing

approved

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Finite Group.</a> From MathWorld--A Wolfram Web Resource.

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Semigroup.html">Semigroup.</a> From MathWorld--A Wolfram Web Resource

approved

editing

reviewed

approved

proposed

reviewed

Number of nonisomorphic semigroups of order n - minus number of groups of order n.

In a sense, this measures the increase in combinatorial structures available by dropping the requirement of inverses, and an identity element, in moving from the group axioms to the semigroup axioms. A semigroup is mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. Other sequences may be derived by considering commutative semigroups and commutative groups, self-converse semigroup, counting idempotents, and the like.

nonn,hard,less,changed

allocated for Jonathan Vos Post Number of nonisomorphic semigroups of order n - number of groups of order n.

0, 4, 23, 186, 1914, 28632, 1627671, 3684030412, 105978177936290

1,2

In a sense, this measures the increase in combinatorial structures available by dropping the requirement of inverses, and an identity element, in moving from the group axioms to the semigroup axioms. A semigroup is mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. Other sequences may be derived by considering commutative semigroups and commutative groups, self-converse semigroup, counting idempotents, and the like.

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Finite Group.</a> From MathWorld--A Wolfram Web Resource.

Eric W. Weisstein, <a href="http://mathworld.wolfram.com/Semigroup.html">Semigroup.</a> From MathWorld--A Wolfram Web Resource

a(n) = A027851(n) - A000001(n).

a(1) = 0 because there are unique groups and semigroups of order 1, so 1 - 1 = 0.

a(2) = 4 because there are 5 semigroups of order 2 groups and a unique group of order 2, so 5 - 1 = 4.

Cf. A000001, A027851, A001423, A029851, A001426, A023814, A058108, A079173, A001329, A186116.

allocated

nonn,hard

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 13 2011

approved

proposed

allocated for Jonathan Vos Post

allocated

approved