semigroup3 http://www.openmath.org/cd http://www.openmath.org/cd/semigroup3.ocd 2006-06-01 2004-06-01 3 1 experimental Semigroup constructions Initiated by Arjeh M. Cohen 2003-10-02 cyclic_semigroup This symbol denotes the cyclic semigroup with a cycle of length l and a tail of length k. The size of cyclic_semigroup(k,l) equals k+l. maps_semigroup This is a unary function whose argument must be a set X or a positive integer. When applied to X, it refers to the semigroup of all functions from X to X if X is a set and to {1,...,X} if X is an integer, whose binary operation is composition of maps and whose identity element is the identity map on the set X, respectively {1,...,X}. left_regular_representation This is a unary function whose argument must be a semigroup M. When applied to M, it represents the map from M to the maps semigroup on M that assigns to m left multiplication by m on M. The left regular representation on M applied to the element x of M represents left multiplication by x on M The left regular representation is a homomorphism of semigroups from M to the maps semigroup on M. automorphism_group This is a function with a single argument which must be a semigroup. It refers to the automorphism group of its argument. direct_product This is an n-ary function whose arguments must be semigroups. It refers to the direct product of its arguments. direct_power This is a binary function whose first argument should be a semigroup M and whose second argument should be a natural number n. It refers to the direct product of n copies of M. free_semigroup This symbol represents a binary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free semigroup generated by the entries of the list or set. The free semigroup on the letters a, b: