%------------------------------------------------------------------------------
% Groupoids definition file
%
%     Author: Rick Butler
%             David Lester, Manchester University & NIA
%
%     Version 1.0            3/1/02
%     Version 1.1           12/3/03   New library structure
%     Version 1.2            5/5/04   Reworked for definition files DRL
%------------------------------------------------------------------------------

groupoid_def[T:Type,*:[T,T->T]]: THEORY

BEGIN

  S: VAR set[T]

  star_closed?(S): bool           = (FORALL (x,y:(S)): member(x*y,S))

  groupoid?(S): MACRO bool        = star_closed?(S)
  finite_groupoid?(S): bool       = groupoid?(S) AND is_finite(S)
  commutative_groupoid?(S): bool  = groupoid?(S) AND commutative?[(S)](*)
  finite_commutative_groupoid?(S): bool
                                  = finite_groupoid?(S) AND
                                    commutative?[(S)](*)

END groupoid_def