Algebra is the study of mathematical manipulations that preserve something (like equality –when solving equations). The areas in which Algebra finds application are quite diverse, from Ancient Greek Geometry through to Modern Information Protection and Security (error correcting codes, data compression, and cryptography). This course begins with topics that should be familiar (such as rulerand-...

A logic may be called local if each model may be used to deene the truth. In the proposition calculus each boolean algebra deene the truth by its greatest element. In the equational logic the equality between the algebra elements deene the truth. In the clause Horn logic the truth is given by the interpretation of the predicats in the model. We give a categorial framework for these logics and w...

This paper describes a new object-oriented model and query algebra that will be used as an input language for the query optimizers that are being built as a part of the EREQ project. The model adopts a uniform view of objects and values and separates syntactic, semantic, and implementation concerns. The algebra addresses issues of type-deened equality and duplicate elimination as well as extens...

We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank l is bounded above by the effective central charge c̃. We show that lattice vertex operator algebras may be characterized by the equalities c̃ = l = c, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator a...

We consider observational specifications of state-based systems which incorporate the declaration of a distinguished set of observer operations. These observers determine an indistinguishability relation for states which is called "observational equality". An important requirement for the non-observer operations is the compatibility with the observational equality. In the CafeOBJ language (and ...

A non-empty set X of a carrier A of an algebra A is called Q-independent if the equality of two term functions f and g of the algebra A on any finite system of elements a1, a2, . . . , an of X implies f(p(a1), p(a2), . . . , p(an)) = g(p(a1), p(a2), . . . , p(an)) for any mapping p ∈ Q. An algebra B is a retract of A if B is the image of a retraction (i.e. of an idempotent endomorphism of B). W...

We consider observational specifications of state-based systems which incorporate the declaration of a distinguished set of observer operations. These observers determine an indistinguishability relation for states which is called "observational equality". An important requirement for the non-observer operations is the compatibility with the observational equality. In the CafeOBJ language (and ...

Abstract Data Types (ADTs) can be specified by the Classified Model (CM) specification language—a first-order Horn language with equality and sort “classification” assertions. Sort assertions generalize the traditional syntactic signatures of ADT specifications, resulting in all of the specification capability of traditional equational specifications, but with the improved expressibility of the...

We construct an infinite dimensional quasi-polyadic equality algebra A such that its cylindric reduct is representable, while A itself is not representable. 1 The most well known generic examples of algebraizations of first order logic are Tarski’s cylindric algebras (CA) and Halmos’ polyadic equality algebras (PEA). The theory of cylindric algebras is well developed in the treatise [10], [11],...

We prove that every place of an algebraic function eld of arbitrary characteristic admits a local uniformization in a nite extension of the function eld. We give a valuation theoretical description of these extensions; in certain cases, they can be found in the henselization of the function eld. For places satisfying the Abhyankar equality and for discrete rational places, no extension is neede...