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E-semigroup
In the area of mathematics known as semigroup theory, an ''E''-semigroup is a semigroup in which the idempotents form a subsemigroup. Certain classes of ''E''-semigroups have been studied long before the more general class, in particular, a regular semigroup that is also an ''E''-semigroup is known as an orthodox semigroup. Weipoltshammer proved that the notion of weak inverse In mathematics, the term weak inverse is used with several meanings. Theory of semigroups In the theory of semigroups, a weak inverse of an element ''x'' in a semigroup is an element ''y'' such that . If every element has a weak inverse, the s ... (the existence of which is one way to define ''E''-inversive semigroups) can also be used to define/characterize ''E''-semigroups as follows: a semigroup ''S'' is an ''E''-semigroup if and only if, for all ''a'' and ''b'' ∈ ''S'', ''W''(''ab'') = ''W''(''b'')''W''(''a''), where ''W''(''x'') ≝ is the set of weak inverses of ''x''. References Semigr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E-inversive Semigroup
__NOTOC__ In abstract algebra, an ''E''-dense semigroup (also called an ''E''-inversive semigroup) is a semigroup in which every element ''a'' has at least one weak inverse In mathematics, the term weak inverse is used with several meanings. Theory of semigroups In the theory of semigroups, a weak inverse of an element ''x'' in a semigroup is an element ''y'' such that . If every element has a weak inverse, the se ... ''x'', meaning that ''xax'' = ''x''. The notion of weak inverse is (as the name suggests) weaker than the notion of inverse used in a regular semigroup (which requires that ''axa''=''a''). The above definition of an ''E''-inversive semigroup ''S'' is equivalent with any of the following: * for every element ''a'' ∈ ''S'' there exists another element ''b'' ∈ ''S'' such that ''ab'' is an idempotent. * for every element ''a'' ∈ ''S'' there exists another element ''c'' ∈ ''S'' such that ''ca'' is an idempotent. This explains the name of the notion as the set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |