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• Semigroup_with_involution abstract "In mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism, which—roughly speaking—brings it closer to a group because this involution, considered as unary operator, exhibits certain fundamental properties of the operation of taking the inverse in a group: uniqueness, double application \"cancelling itself out\", and the same interaction law with the binary operation as in the case of the group inverse. It is thus not a surprise that any group is a semigroup with involution. However, there are significant natural examples of semigroups with involution that are not groups.An example from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law (AB)T = BTAT which has the same form of interaction with multiplication as taking inverses has in the general linear group (which is a subgroup of the full linear monoid). However, for an arbitrary matrix AAT does not equal the identity element (namely the diagonal matrix). Another example, coming from formal language theory, is the free semigroup generated by a nonempty set (an alphabet), with string concatenation as the binary operation, and the involution being the map which reverses the linear order of the letters in a string. A third example, from basic set theory, is the set of all binary relations between a set and itself, with the involution being the inverse relation, and the multiplication given by the usual composition of relations.Semigroups with involution appeared explicitly named in a 1953 paper of Viktor Wagner (in Russian) as result of his attempt to bridge the theory of semigroups with that of semiheaps.".
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• Semigroup_with_involution wikiPageWikiLink *-algebra.
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• Semigroup_with_involution wikiPageWikiLink Abstract_algebra.
• Semigroup_with_involution wikiPageWikiLink Alphabet_(formal_languages).
• Semigroup_with_involution wikiPageWikiLink Antihomomorphism.
• Semigroup_with_involution wikiPageWikiLink Baer_ring.
• Semigroup_with_involution wikiPageWikiLink Band_(mathematics).
• Semigroup_with_involution wikiPageWikiLink Bijection.
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• Semigroup_with_involution wikiPageWikiLink Boris_M._Schein.
• Semigroup_with_involution wikiPageWikiLink Bounded_operator.
• Semigroup_with_involution wikiPageWikiLink C*-algebra.
• Semigroup_with_involution wikiPageWikiLink Category:Algebraic_structures.
• Semigroup_with_involution wikiPageWikiLink Category:Semigroup_theory.
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• Semigroup_with_involution wikiPageWikiLink Concatenation.
• Semigroup_with_involution wikiPageWikiLink Confluence_(abstract_rewriting).
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• Semigroup_with_involution wikiPageWikiLink Function_(mathematics).
• Semigroup_with_involution wikiPageWikiLink Function_composition.
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• Semigroup_with_involution wikiPageWikiLink Greens_relations.
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• Semigroup_with_involution wikiPageWikiLink Harold_Scott_MacDonald_Coxeter.
• Semigroup_with_involution wikiPageWikiLink Heap_(mathematics).
• Semigroup_with_involution wikiPageWikiLink Hermitian_adjoint.
• Semigroup_with_involution wikiPageWikiLink Hermitian_matrix.
• Semigroup_with_involution wikiPageWikiLink Hilbert_space.
• Semigroup_with_involution wikiPageWikiLink Idempotence.
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• Semigroup_with_involution wikiPageWikiLink Identity_function.
• Semigroup_with_involution wikiPageWikiLink Image_(mathematics).
• Semigroup_with_involution wikiPageWikiLink Inclusion_map.
• Semigroup_with_involution wikiPageWikiLink Invariant_(mathematics).
• Semigroup_with_involution wikiPageWikiLink Inverse_element.
• Semigroup_with_involution wikiPageWikiLink Inverse_function.
• Semigroup_with_involution wikiPageWikiLink Inverse_relation.
• Semigroup_with_involution wikiPageWikiLink Inverse_semigroup.
• Semigroup_with_involution wikiPageWikiLink Involution_(mathematics).
• Semigroup_with_involution wikiPageWikiLink Isomorphism.
• Semigroup_with_involution wikiPageWikiLink Linear_algebra.
• Semigroup_with_involution wikiPageWikiLink Map_(mathematics).
• Semigroup_with_involution wikiPageWikiLink Mathematics.
• Semigroup_with_involution wikiPageWikiLink Matrix_(mathematics).
• Semigroup_with_involution wikiPageWikiLink Matrix_multiplication.
• Semigroup_with_involution wikiPageWikiLink Michael_P._Drazin.
• Semigroup_with_involution wikiPageWikiLink Monoid.
• Semigroup_with_involution wikiPageWikiLink Moore–Penrose_pseudoinverse.
• Semigroup_with_involution wikiPageWikiLink Ordered_groupoid.
• Semigroup_with_involution wikiPageWikiLink Orthogonal_complement.
• Semigroup_with_involution wikiPageWikiLink Partial_equivalence_relation.
• Semigroup_with_involution wikiPageWikiLink Partial_groupoid.
• Semigroup_with_involution wikiPageWikiLink Partial_isometry.
• Semigroup_with_involution wikiPageWikiLink Partially_ordered_set.
• Semigroup_with_involution wikiPageWikiLink Piecewise.
• Semigroup_with_involution wikiPageWikiLink Quantum_mechanics.
• Semigroup_with_involution wikiPageWikiLink Quotient_algebra.
• Semigroup_with_involution wikiPageWikiLink Real_number.
• Semigroup_with_involution wikiPageWikiLink Regular_semigroup.
• Semigroup_with_involution wikiPageWikiLink Rewriting.
• Semigroup_with_involution wikiPageWikiLink Semigroup.
• Semigroup_with_involution wikiPageWikiLink Semigroup_Forum.
• Semigroup_with_involution wikiPageWikiLink Semilattice.
• Semigroup_with_involution wikiPageWikiLink Set_theory.
• Semigroup_with_involution wikiPageWikiLink Special_classes_of_semigroups.
• Semigroup_with_involution wikiPageWikiLink String_(computer_science).
• Semigroup_with_involution wikiPageWikiLink Total_order.
• Semigroup_with_involution wikiPageWikiLink Transpose.
• Semigroup_with_involution wikiPageWikiLink Unary_operation.
• Semigroup_with_involution wikiPageWikiLink Universal_algebra.

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