examples of algebraic systems
Selected examples of algebraic systems are specified below.
A set is an algebra where .
A monoid is an algebra of type . However, not every algebra of type is a monoid.
A quandle is an algebraic system of type . It has the same type as a lattice.
An -group (http://planetmath.org/PolyadicSemigroup) is an algebraic system of type .
The set of all well-formed formulas over a set of propositional variables can be thought of as an algebraic system, as each of the logical connectives as an operation on may be associated with a finitary operation on . In classical propositional logic, the algebraic system may be of type , if we consider and as the only logical connectives; or it may be of type , if the full set is used.
Below are some non-examples of algebraic systems:
A complete lattice is not, in general, an algebraic system because the arbitrary meet and join operations are not finitary.
A field is not an algebraic system, since, in addition to the five operations of a ring, there is the multiplicative inverse operation, which is not defined for .
|Title||examples of algebraic systems|
|Date of creation||2013-03-22 18:40:11|
|Last modified on||2013-03-22 18:40:11|
|Last modified by||CWoo (3771)|