Duality (order theory)

In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by P^op or P^d. This dual order P^op is defined to be the set with the inverse order, i.e. x ≤ y holds in P^opif and only if y ≤ x holds in P. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two posets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other.

The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets:

If a statement or definition is equivalent to its dual then it is said to be self-dual. Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol.

Chess theory

The game of chess is commonly divided into three phases: the opening, middlegame, and endgame. There is a large body of theory regarding how the game should be played in each of these phases, especially the opening and endgame. Those who write about chess theory, who are often but not necessarily also eminent players, are referred to as "theorists" or "theoreticians".

"Opening theory" commonly refers to consensus, broadly represented by current literature on the openings. "Endgame theory" consists of statements regarding specific positions, or positions of a similar type, though there are few universally applicable principles. "Middlegame theory" often refers to maxims or principles applicable to the middlegame. The modern trend, however, is to assign paramount importance to analysis of the specific position at hand rather than to general principles.

The development of theory in all of these areas has been assisted by the vast literature on the game. In 1913, preeminent chess historian H. J. R. Murray wrote in his 900-page magnum opus A History of Chess that, "The game possesses a literature which in contents probably exceeds that of all other games combined." He estimated that at that time the "total number of books on chess, chess magazines, and newspapers devoting space regularly to the game probably exceeds 5,000". In 1949, B. H. Wood opined that the number had increased to about 20,000.David Hooper and Kenneth Whyld wrote in 1992 that, "Since then there has been a steady increase year by year of the number of new chess publications. No one knows how many have been printed..." The world's largest chess library, the John G. White Collection at the Cleveland Public Library, contains over 32,000 chess books and serials, including over 6,000 bound volumes of chess periodicals. Chess players today also avail themselves of computer-based sources of information.

Scientific theory

A scientific theory is a well-substantiated explanation of some aspect of the natural world that is acquired through the scientific method and repeatedly tested and confirmed through observation and experimentation. As with most (if not all) forms of scientific knowledge, scientific theories are inductive in nature and aim for predictive power and explanatory capability.

The strength of a scientific theory is related to the diversity of phenomena it can explain, and to its elegance and simplicity. See Occam's razor. As additional scientific evidence is gathered, a scientific theory may be rejected or modified if it does not fit the new empirical findings; in such circumstances, a more accurate theory is then desired. In certain cases, the less-accurate unmodified scientific theory can still be treated as a theory if it is useful (due to its sheer simplicity) as an approximation under specific conditions (e.g., Newton's laws of motion as an approximation to special relativity at velocities which are small relative to the speed of light).

Order and disorder (physics)

In physics, the terms order and disorder designate the presence or absence of some symmetry or correlation in a many-particle system.

In condensed matter physics, systems typically are ordered at low temperatures; upon heating, they undergo one or several phase transitions into less ordered states. Examples for such an order-disorder transition are:

The degree of freedom that is ordered or disordered can be translational (crystalline ordering), rotational (ferroelectric ordering), or a spin state (magnetic ordering).

The order can consist either in a full crystalline space group symmetry, or in a correlation. Depending on how the correlations decay with distance, one speaks of long-range order or short-range order.

If a disordered state is not in thermodynamic equilibrium, one speaks of quenched disorder. For instance, a glass is obtained by quenching (supercooling) a liquid. By extension, other quenched states are called spin glass, orientational glass. In some contexts, the opposite of quenched disorder is annealed disorder.

Fictional universe of The Guardians of Time

In the fictional Guardians of Time Trilogy, author Marianne Curley constructs a highly detailed alternate universe in which much of the action of the series takes place, unknown to the regular world. In this universe, two organizations battle for control over time.

The Guard

The Guardians of Time (known also as The Guard) is a society dedicated to preserving history against the attempts of the Order of Chaos to alter it. It is headed by a sexless immortal called Lorian, who is backed by a Tribunal of nine members, each a representative of a house.

The headquarters of the Tribunal, as well as the Guard itself, is located in Athens, year 200 BC, outside of the mortal measurements of time. For their purposes, they also use a place called the Citadel, connected to another area known as the labyrinth (also used by Order of Chaos) which serves as a disembarkation point for the Guards' missions into the past. Guard meetings frequently take place in Arkarian's, a Guard member's, abode hidden within the depths of a mountain. Connected to this mountain is the hidden city of Veridian and later learned, also connected to Neriah's fortress.

Order (business)

In business or commerce, an order is a stated intention, either spoken or written, to engage in a commercial transaction for specific products or services. From a buyer's point of view it expresses the intention to buy and is called a purchase order. From a seller's point of view it expresses the intention to sell and is referred to as a sales order. When the purchase order of the buyer and the sales order of the seller agree, the orders become a contract between the buyer and seller.

Within an organization, the term order may be used to refer to a work order for manufacturing, a preventive maintenance order, or an order to make repairs to a facility.

In many businesses, orders are used to collect and report costs and revenues according to well-defined purposes. Then it is possible to show for what purposes costs have been incurred.

Spoken orders

Businesses such as retail stores, restaurants and filling stations conduct business with their customers by accepting orders that are spoken or implied by the buyer's actions. Taking a shopping cart of merchandise to a check-out counter is an implied intent to buy the merchandise. Placing a take-out or eat-in order at a restaurant is a spoken purchase order. Putting gasoline in one's tank at a filling station is an implied order. The seller usually expects immediate payment by cash, check or credit card for these purchases, and the seller provides the buyer with a receipt for the payment. In legal terms, this form of business order is an "implied in fact contract".

Duality (mechanical engineering)

In mechanical engineering, many terms are associated into pairs called duals. A dual of a relationship is formed by interchanging force (stress) and deformation (strain) in an expression.

Here is a partial list of mechanical dualities:

Examples

Constitutive relation

See also

References