|
Epigroup
In abstract algebra, an epigroup is a semigroup in which every element has a power that belongs to a subgroup. Formally, for all ''x'' in a semigroup ''S'', there exists a positive integer ''n'' and a subgroup ''G'' of ''S'' such that ''x''''n'' belongs to ''G''. Epigroups are known by wide variety of other names, including quasi-periodic semigroup, group-bound semigroup, completely π-regular semigroup, strongly π-regular semigroup (sπr), or just π-regular semigroup (although the latter is ambiguous). More generally, in an arbitrary semigroup an element is called ''group-bound'' if it has a power that belongs to a subgroup. Epigroups have applications to ring theory. Many of their properties are studied in this context. Epigroups were first studied by Douglas Munn in 1961, who called them ''pseudoinvertible''. Properties * Epigroups are a generalization of periodic semigroups, thus all finite semigroups are also epigroups. * The class of epigroups also contains all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely 0-simple Semigroups
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ''ab'' = ''ba'' for all elements ''a'' and ''b'' in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively. In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Classes Of Semigroups
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ''ab'' = ''ba'' for all elements ''a'' and ''b'' in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively. In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eventually Regular Semigroup
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ''ab'' = ''ba'' for all elements ''a'' and ''b'' in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively. In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Relations
In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951. John Mackintosh Howie, a prominent semigroup theorist, described this work as "so all-pervading that, on encountering a new semigroup, almost the first question one asks is 'What are the Green relations like?'" (Howie 2002). The relations are useful for understanding the nature of divisibility in a semigroup; they are also valid for groups, but in this case tell us nothing useful, because groups always have divisibility. Instead of working directly with a semigroup ''S'', it is convenient to define Green's relations over the monoid ''S''1. (''S''1 is "''S'' with an identity adjoined if necessary"; if ''S'' is not already a monoid, a new element is adjoined and defined to be an identity.) This ensures that principal ideals generated by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nambooripad Order
In mathematics, Nambooripad order (also called Nambooripad's partial order) is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig, some authors refer to it as Hartwig–Nambooripad order. "Natural" here means that the order is defined in terms of the operation on the semigroup. In general Nambooripad's order in a regular semigroup is not compatible with multiplication. It is compatible with multiplication only if the semigroup is pseudo-inverse (locally inverse). Precursors Nambooripad's partial order is a generalisation of an earlier known partial order on the set of idempotents in any semigroup. The partial order on the set ''E'' of idempotents in a semigroup ''S'' is defined as follows: For any ''e'' and ''f'' in ''E'', ''e'' ≤ ''f'' if and only if ''e'' = ''ef'' = ''fe''. Vagner in 1952 had extended t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Semigroup
In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups. Structure The monogenic semigroup generated by the singleton set is denoted by \langle a \rangle . The set of elements of \langle a \rangle is . There are two possibilities for the monogenic semigroup \langle a \rangle : * ''a'' ''m'' = ''a'' ''n'' ⇒ ''m'' = ''n''. * There exist ''m'' ≠ ''n'' such that ''a'' ''m'' = ''a'' ''n''. In the former case \langle a \rangle is isomorphic to the semigroup ( , + ) of natural numbers under addition. In such a case, \langle a \rangle is an ''infinite monogenic semigroup'' and the element ''a'' is said to have ''infinite order''. It is sometimes called the ''free monogenic semigroup'' because it is also a free semigroup with one generator. In the latter case let ''m'' be the smallest positive integer such that ''a'' ''m'' = ''a'' ''x'' for some positive integer ''x'' ≠ ''m'', and let ''r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cancellative Semigroup
In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form ''a''·''b'' = ''a''·''c'', where · is a binary operation, one can cancel the element ''a'' and deduce the equality ''b'' = ''c''. In this case the element being cancelled out is appearing as the left factors of ''a''·''b'' and ''a''·''c'' and hence it is a case of the left cancellation property. The right cancellation property can be defined analogously. Prototypical examples of cancellative semigroups are the positive integers under addition or multiplication. Cancellative semigroups are considered to be very close to being groups because cancellability is one of the necessary conditions for a semigroup to be embeddable in a group. Moreover, every finite cancellative semigroup is a group. One of the main problems associated with the study of cancellative sem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Partial Order
In mathematics, Nambooripad order (also called Nambooripad's partial order) is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig, some authors refer to it as Hartwig–Nambooripad order. "Natural" here means that the order is defined in terms of the operation on the semigroup. In general Nambooripad's order in a regular semigroup is not compatible with multiplication. It is compatible with multiplication only if the semigroup is pseudo-inverse (locally inverse). Precursors Nambooripad's partial order is a generalisation of an earlier known partial order on the set of idempotents in any semigroup. The partial order on the set ''E'' of idempotents in a semigroup ''S'' is defined as follows: For any ''e'' and ''f'' in ''E'', ''e'' ≤ ''f'' if and only if ''e'' = ''ef'' = ''fe''. Vagner in 1952 had extended t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Extension
Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered in abstract algebra * Ideal, special subsets of a semigroup * Ideal (order theory), special kind of lower sets of an order * Ideal (set theory), a collection of sets regarded as "small" or "negligible" * Ideal (Lie algebra), a particular subset in a Lie algebra * Ideal point, a boundary point in hyperbolic geometry * Ideal triangle, a triangle in hyperbolic geometry whose vertices are ideal points Science * Ideal chain, in science, the simplest model describing a polymer * Ideal gas law, in physics, governing the pressure of an ideal gas * Ideal transformer, an electrical transformer having zero resistance and perfect magnetic threading * Ideal final result, in TRIZ methodology, the best possible solution * Thought experiment, sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brandt Semigroup
In mathematics, Brandt Brandt may refer to: Places United States * Brandt, Ohio, an unincorporated community * Brandt, South Dakota, a town * Brandt Township, Polk County, Minnesota Elsewhere * Mount Brandt, Queen Maud Land, Antarctica * Brandt Cove, South Georgia Is ... semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups: Let ''G'' be a group and I, J be non-empty sets. Define a matrix P of dimension , I, \times , J, with entries in G^0=G \cup \. Then, it can be shown that every 0-simple semigroup is of the form S=(I\times G^0\times J) with the operation (i,a,j)*(k,b,n)=(i,a p_ b,n). As Brandt semigroups are also inverse semigroups, the construction is more specialized and in fact, I = J (Howie 1995). Thus, a Brandt semigroup has the form S=(I\times G^0\times I) with the operation (i,a,j)*(k,b,n)=(i,a p_ b,n) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + '' potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid (\mathbb, \times) of the natural numbers with multiplication, on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> Idempotent")