Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations, and relations that are defined on it.

Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols.

Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model, if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a semantic model when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as interpretations.

In database theory, structures with no functions are studied as models for relational databases, in the form of relational models.

Structure (category theory)

In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications. In fact this is a normal way of proceeding; in the absence of recognisable structure (which might be hidden) problems tend to fall into the combinatorics classification of matters requiring special arguments.

In category theory structure is discussed implicitly - as opposed to the explicit discussion typical with the many algebraic structures. Starting with a given class of algebraic structure, such as groups, one can build the category in which the objects are groups and the morphisms are group homomorphisms: that is, of structures on one type, and mappings respecting that structure. Starting with a category C given abstractly, the challenge is to infer what structure it is on the objects that the morphisms 'preserve'.

The term structure was much used in connection with the Bourbaki group's approach. There is even a definition. Structure must definitely include topological space as well as the standard abstract algebra notions. Structure in this sense is probably commensurate with the idea of concrete category that can be presented in a definite way - the topological case means that infinitary operations will be needed. Presentation of a category (analogously to presentation of a group) can in fact be approached in a number of ways, the category structure not being (quite) an algebraic structure in its own right.

Structure (disambiguation)

The structure of a thing is how the parts of it relate to each other, how it is "assembled".

Structure may also refer to:

In architecture:

In engineering:

In art:

In biology:

In chemistry:

Face (album)

Face is the debut album of Of Cabbages and Kings, released by Purge/Sound League in 1988.

Track listing

All music composed by Of Cabbages and Kings.

Personnel

Adapted from the Face liner notes.

Release history

References

Face (sociological concept)

The term face idiomatically refers to one's own sense of dignity or prestige in social contexts. In the English-speaking world, the expression "To save face" describes the lengths that an individual may go to in order to preserve their established position in society, taking action to ensure that one is not thought badly of by their peers. It is a fundamental concept in the fields of sociology, sociolinguistics, semantics, politeness theory, psychology, political science, communication, and face negotiation theory, and translates at least somewhat equivalently into many world languages, both Germanic and otherwise.

Definitions

Although Lin Yutang claimed "Face cannot be translated or defined", compare these definitions:

Face (a cappella group)

Face are a Boulder, Colorado-based "all-vocal rock band," or a cappella group performing mostly rock music. National appearances and awards include one of the eight original groups on NBC's premiere season of "The Sing-Off" (2009), two-time Runner-Up and two-time National Audience Favorite at the National Finals of Harmony Sweepstakes A Cappella Festival (2005 and 2007), and Runner-Up for Favorite Pop/Rock Group from the Contemporary A Cappella Society's Community Awards (2007). Based in Boulder, Colorado, Face has also garnered numerous local awards including Best Local Band by The Denver Channel's A-List Awards (2013), Best Local Musician by Boulder Weekly's Best of Boulder Awards (2009) and third-place for Best Local Band by The Denver Channel's A-List Awards (2009).

History

The idea for Face grew out of a university a cappella experience. Both Ben Lunstad and Joseph DiMasi had formed and performed with undergraduate a cappella groups. They met in grad school in 2000 at the University of Colorado in Boulder while singing with CU's In The Buff. Deciding that In The Buff wasn't exactly what they were looking for, Ben and DiMasi co-founded the award-winning Extreme Measures.