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In full generality, an algebraic structure may use any number of sets and any number of axioms in its definition. The most commonly studied structures, however, usually involve only one or two sets and one or two binary operations. The structures below are organized by how many sets are involved, and how many binary operations are used. Increased indentation is meant to indicated a more exotic structure, and the least indented levels are the most basic. One binary operation on one set Group-like structures Totality* Associativity Identity Divisibility Commutativity Magma Yes No No No No Semigroup Yes Yes No No No Monoid Yes Yes Yes No No Group Yes Yes Yes Yes No Abelian Group Yes Yes Yes Yes Yes Loop Yes No Yes Yes No Quasigroup Yes No No Yes No Groupoid No Yes Yes Yes No Category No Yes Yes No No Semicategory No Yes No No No *Closure, which is used in many sources to define group-like structures, is an equivalent axiom to totality, though defined differently. The following structures consist of a set with a binary operation. The most common structure is that of a group. Other structures involve weakening or strengthening the axioms for groups, and may additionally use unary operations. Groups are key structures. Abelian groups are an important special type of group. semigroups and monoids: These are like groups, except the operation need not have inverse elements. quasigroups and loops: These are like groups, except the operation need not be associative. Magmas: These are like groups, except the operation need not be associative or have inverse elements. Semilattice: This is basically "half" of a lattice structure (see below). Two binary operations on one set The main types of structures with one set having two binary operations are rings and lattices. The axioms defining many of the other structures are modifications of the axioms for rings and lattices. One major difference between rings and lattices is that their two operations are related to each other in different ways. In ring-like structures, the two operations are linked by the distributive law; in lattice-like structures, the operations are linked by the absorption law. Rings: The two operations are usually called addition and multiplication. Commutative rings are an especially important type of ring where the multiplication operation is commutative. Integral domains and fields are especially important types of commutative rings. Nonassociative rings: These are like rings, but the multiplication operation need not be associative. Lie rings and Jordan rings are special examples of nonassociative rings. semirings: These are like rings, but the addition operation need not have inverses. nearrings: These are like rings, but the addition operation need not be commutative. *-rings: These are rings with an additional unary operation known as an involution. Lattices: The two operations are usually called meet and join. Latticoid: meet and join commute but need not associate. Skew lattice: meet and join associate but need not commute. Two binary operations and two sets The following structures have the common feature of having two sets, A and B, so that there is a binary operation from A×A into A and another operation from A×B into A. Vector spaces: The set A is an Abelian group, and the set B is a field. Graded vector spaces: Vector spaces which are equipped with a direct sum decomposition into subspaces. Modules: The set A is an Abelian group, but the B is only a general ring and not necessarily a field. Special types of modules, including free modules, projective modules, injective modules and flat modules are studied in abstract algebra. Group with operators: In this case, the set A is a group, and the set B is just a set. Three binary operations and two sets Many structures here are actually hybrid structures of the previously mentioned ones. Algebra over a field: This is a ring which is also a vector space over a field. There are axioms governing the interaction of the two structures. Multiplication is usually assumed to be associative. Algebra over a ring: These are defined the same way as algebras over fields, except that the field may now be any commutative ring. Graded algebra: These algebras are equipped with a decomposition into grades. Non-associative algebras: These are algebras for which the associativity of ring multiplication is relaxed. Lie algebras and Jordan algebras are special examples of non-associative algebras. Coalgebra: This structure has axioms which make its multiplication dual to those of an associative algebra. Bialgebra: These structures are simultaneously algebras and coalgebras whose operations are compatible. There are actually four operations for this structure. |
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