Categories as algebra: An essential ingredient in the theory of monoids

摘要: Abstract The study of categories as generalized monoids is shown to be essential the understanding monoid decomposition theory. A new ordering for categories, called division, introduced which allows useful comparison and categories. Associated with every morphism ϕ : M → N a category D , derived ϕ, encodes information about morphism. rightful generalization kernel group Collections that admit direct products division are studied. Called varieties, these collections completely determined by path equations their members satisfy. Several important examples discussed. In major application, strong connection established between varieties certain semigroup associated recognizable languages.