Ordered algebraic structures

In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S.

In mathematics, a real closed ring (RCR) is a commutative ring A that is a subring of a product of real closed fields, which is closed under continuous semi-algebraic functions defined over the integers.

In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Such fields will contain infinitesimal and infinitely large elements, suitably defined.

In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order ≤ {\\displaystyle \\,\\leq \\,} on the underlying set A that is compatible with the ring operations in the sense that it satisfies: x ≤ y  implies  x + z ≤ y + z {\\displaystyle x\\leq y{\\text{ implies }}x+z\\leq y+z} and 0 ≤ x  and  0 ≤ y  imply that  0 ≤ x ⋅ y {\\displaystyle 0\\leq x{\\text{ and }}0\\leq y{\\text{ imply that }}0\\leq x\\cdot y} for all x , y , z ∈ A {\\displaystyle x,y,z\\in A} . Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring ( A , ≤ ) {\\displaystyle (A,\\leq )} where partially ordered additive group is Archimedean.

In abstract algebra, a partially ordered group is a group (G, +) equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then a + g ≤ b + g and g + a ≤ g + b.

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard orderings.

In mathematics, biquandles and biracks are sets with binary operations that generalize quandles and racks. Biquandles take, in the theory of virtual knots, the place that quandles occupy in the theory of classical knots. Biracks and racks have the same relation, while a biquandle is a birack which satisfies some additional conditions.

In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations coincide when the monoid is commutative. The general concept was introduced by Morgan Ward and Robert P. Dilworth in 1939. Examples, some of which existed prior to the general concept, include Boolean algebras, Heyting algebras, residuated Boolean algebras, relation algebras, and MV-algebras. Residuated semilattices omit the meet operation ∧, for example Kleene algebras and action algebras.