monoid3 http://www.openmath.org/cd/monoid3.ocd 2006-06-01 2004-06-01 3 2 experimental Monoid constructions Initiated by Arjeh M. Cohen 2003-10-02 Edited AMC 2004-03-05 Edited AMC 2004-06-27 cyclic_monoid This symbol is a function of two natural numbers, the first of which should be positive. When evaluated at k and l, it denotes the cyclic monoid with a cycle of length l and a tail (including the identity element) of length k. The size of cyclickl(k,l) equals k+l. maps_monoid This is a unary function whose argument must be a set X or a positive integer. When applied to X, it refers to the monoid 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 monoid M. When applied to M, it represents the map from M to the maps monoid 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 monoids from M to the maps monoid on M. automorphism_group This is a function with a single argument which must be a monoid. It refers to the automorphism group of its argument. direct_product This is an n-ary function whose arguments must be monoids. It refers to the direct product of its arguments. direct_power This is a binary function whose first argument should be a monoid M and whose second argument should be a natural number n. It refers to the direct product of n copies of M. free_monoid This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the free monoid generated by the entries of the list or set. The free monoid on the letters a, b: strings This symbol represents a unary function. The argument is a list or a set. When evaluated on such an argument, the function represents the set of all strings whose characters are entries of the list or set. emptyword This symbol represents a constant. It represents the empty string. concatenation This symbol represents a binary concatenation operation on strings.