.Open One Associative Operator A set of elements with a single operation on it is one of the commonest structures on modern mathematics -- especially when the operation is associative: a * (b*c) = (a*b)*c. In \$MATHS we document three forms depending on whether the operation has a unit: a *1 = 1* a = a. Further, whether it has an inverse a*x=x*a = 1. . Semigroup A set of elements that have an associative operator is called a semigroup. The formal structure is defined as: SEMIGROUP::=Net{Set:Sets, op:\$associative(Set)}. Detailed properties are documented in .See ./math_32_Semigroups.html We also use the same word to refer to the set of elements, when there exists such an operation: semigroup::={ Set:Sets || (Set,op) in \$ \$SEMIGROUP}, The same word (semigroup) with different capitalization indicates the collection of tpls (Set, op) that form semigroups: Semigroup::=\$ \$SEMIGROUP. ()|- (A,B) in Semigroup iff A in Sets and op in \$associative(A). It is also helpful to have the following abreviations: For op, semigroup(op) ::={ Set:Sets || (Set,op) in \$ \$SEMIGROUP}, For *:infix(T1), semigroup(*) ::={S:@T1||SEMIGROUP(Set=>S,op=>*)} The above set of definitions is typical of the various ways the name of an algebraic system can be used. This package defining `semigroup` and `Semigroup` in terms of `SEMIGROUP` can be quickly introduced by asserting: |- \$ALGEBRA(\$SEMIGROUP, Semigroup, semigroup). A semigroup is said to be Abelian if its operator commutes. ABELIAN::=Net{Set:Sets, op:\$commutative(Set)}. ABELIAN_SEMIGROUP::=\$SEMIGROUP and \$ABELIAN. Abelian_semigroup::=\$ \$ABELIAN_SEMIGROUP. Abelian semigroups are also called COMMUTATIVE_SEMIGROUPS::=http://www.csci.csusb.edu/dick/maths/math_32_Semigroups.html#Commutative_Semigroups. A lattice is actually two semilattices - where a semilattice is a semigroup with an idempotent operation: SEMILATTICE::=\$SEMIGROUP and \$ABELIAN and Net{op in \$idempotents(S)}. |- \$ALGEBRA(\$SEMILATTICE, Semilattice, semilattice). For more on semigroups see .See http://www/dick/maths/math_32_Semigroups.html . Monoid A monoid is a semigroup with a unit. See MONOIDS::=http://www/dick/maths/math_33_Monoids.html for detailed results. (MONOIDS)|- MONOID::=\$SEMIGROUP and Net{ u:\$units(Set,op). ...}. |-\$ALGEBRA(\$MONOID, Monoid, monoid). . Group A group is a monoid with an invertable operation. .See http://www/dick/maths/math_34_Groups.html ()|- GROUP::=\$MONOID and Net{ i:Set->Set, for all x:Set(x op x.i= x.i op x =u). ...}. |- \$ALGEBRA(GROUP, Group, group). .Close One Associative Operator . Links associative::=http://www/dick/maths/math_11_STANDARD.html#associative. commutative::=http://www/dick/maths/math_11_STANDARD.html#commutative. units::=http://www/dick/maths/math_11_STANDARD.html#units. zeroes::=http://www/dick/maths/math_11_STANDARD.html#zeroes. idempotents::=http://www/dick/maths/math_11_STANDARD.html#idempotents. ALGEBRA::=http://www.csci.csusb.edu/dick/maths/math_43_Algebras.html#ALGEBRA.