Dimension theory (algebra)In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety).
Hyperbolic geometryIn mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point.
Ideal (order theory)In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory. A subset I of a partially ordered set is an ideal, if the following conditions hold: I is non-empty, for every x in I and y in P, y ≤ x implies that y is in I (I is a lower set), for every x, y in I, there is some element z in I, such that x ≤ z and y ≤ z (I is a directed set).
Principal idealIn mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in The remainder of this article addresses the ring-theoretic concept.
Boolean prime ideal theoremIn mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article focuses on prime ideal theorems from order theory.
Trade blocA trade bloc is a type of intergovernmental agreement, often part of a regional intergovernmental organization, where barriers to trade (tariffs and others) are reduced or eliminated among the participating states. Trade blocs can be stand-alone agreements between several states (such as the USMCA) or part of a regional organization (such as the European Union). Depending on the level of economic integration, trade blocs can be classified as preferential trading areas, free-trade areas, customs unions, common markets, or economic and monetary unions.
Economic liberalizationEconomic liberalization, or economic liberalisation, is the lessening of government regulations and restrictions in an economy in exchange for greater participation by private entities. In politics, the doctrine is associated with classical liberalism and neoliberalism. Liberalization in short is "the removal of controls" to encourage economic development. Many countries have pursued and followed the path of economic liberalization in the 1980s, 1990s and in the 21st century, with the stated goal of maintaining or increasing their competitiveness as business environments.
Structural adjustmentStructural adjustment programs (SAPs) consist of loans (structural adjustment loans; SALs) provided by the International Monetary Fund (IMF) and the World Bank (WB) to countries that experience economic crises. Their stated purpose is to adjust the country's economic structure, improve international competitiveness, and restore its balance of payments. The IMF and World Bank (two Bretton Woods institutions) require borrowing countries to implement certain policies in order to obtain new loans (or to lower interest rates on existing ones).
CodimensionIn mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Codimension is a relative concept: it is only defined for one object inside another.
