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• Monoid abstract "In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single object. Thus, they capture the idea of function composition within a set. Monoids are also commonly used in computer science, both in its foundational aspects and in practical programming. The set of strings built from a given set of characters is a free monoid. The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history monoids provide a foundation for process calculi and concurrent computing. Some of the more important results in the study of monoids are the Krohn–Rhodes theorem and the star height problem. The history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups.".
• Monoid wikiPageID "19652".
• Monoid wikiPageLength "29616".
• Monoid wikiPageOutDegree "161".
• Monoid wikiPageRevisionID "678738475".
• Monoid wikiPageWikiLink 0_(number).
• Monoid wikiPageWikiLink 1_(number).
• Monoid wikiPageWikiLink Abelian_group.
• Monoid wikiPageWikiLink Absorbing_element.
• Monoid wikiPageWikiLink Abstract_algebra.
• Monoid wikiPageWikiLink Abstract_data_type.
• Monoid wikiPageWikiLink Abstract_data_types.
• Monoid wikiPageWikiLink Algebra_over_a_field.
• Monoid wikiPageWikiLink Algebraic_structure.
• Monoid wikiPageWikiLink Associative.
• Monoid wikiPageWikiLink Associative_property.
• Monoid wikiPageWikiLink Associativity.
• Monoid wikiPageWikiLink Bicyclic_monoid.
• Monoid wikiPageWikiLink Bicyclic_semigroup.
• Monoid wikiPageWikiLink Bijection.
• Monoid wikiPageWikiLink Bijective.
• Monoid wikiPageWikiLink Binary_operation.
• Monoid wikiPageWikiLink Binary_relation.
• Monoid wikiPageWikiLink Binary_tree.
• Monoid wikiPageWikiLink Boolean_algebra_(structure).
• Monoid wikiPageWikiLink Cambridge_University_Press.
• Monoid wikiPageWikiLink Cancellation_property.
• Monoid wikiPageWikiLink Cancellative.
• Monoid wikiPageWikiLink Cartesian_product.
• Monoid wikiPageWikiLink Category:Algebraic_structures.
• Monoid wikiPageWikiLink Category:Category_theory.
• Monoid wikiPageWikiLink Category:Semigroup_theory.
• Monoid wikiPageWikiLink Category_(mathematics).
• Monoid wikiPageWikiLink Category_of_groups.
• Monoid wikiPageWikiLink Category_of_magmas.
• Monoid wikiPageWikiLink Category_of_monoids.
• Monoid wikiPageWikiLink Category_of_semigroups.
• Monoid wikiPageWikiLink Category_of_sets.
• Monoid wikiPageWikiLink Category_theory.
• Monoid wikiPageWikiLink Character_(computing).
• Monoid wikiPageWikiLink Class_(set_theory).
• Monoid wikiPageWikiLink Closed_manifold.
• Monoid wikiPageWikiLink Closure_operator.
• Monoid wikiPageWikiLink Commutative.
• Monoid wikiPageWikiLink Commutative_monoid.
• Monoid wikiPageWikiLink Commutative_property.
• Monoid wikiPageWikiLink Compact_surface.
• Monoid wikiPageWikiLink Complex_number.
• Monoid wikiPageWikiLink Computer_science.
• Monoid wikiPageWikiLink Concatenation.
• Monoid wikiPageWikiLink Concurrent_computation.
• Monoid wikiPageWikiLink Concurrent_computing.
• Monoid wikiPageWikiLink Connected_sum.
• Monoid wikiPageWikiLink Constant_function.
• Monoid wikiPageWikiLink Cyclic_group.
• Monoid wikiPageWikiLink Data_structure.
• Monoid wikiPageWikiLink Directed_set.
• Monoid wikiPageWikiLink Empty_string.
• Monoid wikiPageWikiLink Endomorphism.
• Monoid wikiPageWikiLink Equivalence_of_categories.
• Monoid wikiPageWikiLink Exclusive_disjunction.
• Monoid wikiPageWikiLink Exclusive_or.
• Monoid wikiPageWikiLink Finitary_closure_operator.
• Monoid wikiPageWikiLink Finite-state_machine.
• Monoid wikiPageWikiLink Finite_state_machines.
• Monoid wikiPageWikiLink Fold_(higher-order_function).
• Monoid wikiPageWikiLink Free_functor.
• Monoid wikiPageWikiLink Free_monoid.
• Monoid wikiPageWikiLink Free_object.
• Monoid wikiPageWikiLink Full_transformation_monoid.
• Monoid wikiPageWikiLink Function_composition.
• Monoid wikiPageWikiLink Functor.
• Monoid wikiPageWikiLink Greens_relations.
• Monoid wikiPageWikiLink Grothendieck_group.
• Monoid wikiPageWikiLink Group_(mathematics).
• Monoid wikiPageWikiLink Group_action.
• Monoid wikiPageWikiLink Group_homomorphism.
• Monoid wikiPageWikiLink Group_presentation.
• Monoid wikiPageWikiLink Heyting_algebra.
• Monoid wikiPageWikiLink History_monoid.
• Monoid wikiPageWikiLink Homeomorphism.
• Monoid wikiPageWikiLink Homomorphism.
• Monoid wikiPageWikiLink Idempotence.
• Monoid wikiPageWikiLink Idempotent.
• Monoid wikiPageWikiLink Identity_element.
• Monoid wikiPageWikiLink Identity_function.
• Monoid wikiPageWikiLink Index_set.
• Monoid wikiPageWikiLink Infimum_and_supremum.
• Monoid wikiPageWikiLink Infinitary_sum.
• Monoid wikiPageWikiLink Integer.
• Monoid wikiPageWikiLink Inverse_element.
• Monoid wikiPageWikiLink Isomorphism.
• Monoid wikiPageWikiLink Join_and_meet.
• Monoid wikiPageWikiLink Kleene_algebra.
• Monoid wikiPageWikiLink Krohn–Rhodes_theorem.
• Monoid wikiPageWikiLink Krohn–Rhodes_theory.
• Monoid wikiPageWikiLink Lattice_(order).
• Monoid wikiPageWikiLink Least_upper_bound.
• Monoid wikiPageWikiLink Left_zero_semigroup.
• Monoid wikiPageWikiLink Lexicographical_order.

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