Mapping Extension of an Algebra

A new construction of algebras called a mapping extension of an algebra is here introduced. The construction yields a generalization of some classical constructions such as the nilpotent extension of an algebra, inflation of a semigroup but also the square extension construction introduced recently for idempotent groupoids. The mapping extension construction is defined for algebras of any fixed type, however nullary operation symbols are here not admitted. It is based on the notion of a retraction and some system of mappings. A mapping extension of a given algebra is constructed as a counterimage algebra by a specially defined retraction. Varieties of algebras satisfying an identity φ(x) ≈ x for a term φ not being a variable (such as varieties of lattices, Boolean algebras, groups and rings) are especially interesting because for such a variety [Formula: see text], all mapping extensions by φ of algebras from [Formula: see text] form an equational class. In the last section, combinatorial properties of the mapping extension construction are considered.

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Injectivity in Equational Classes of Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-017-8 ◽

1972 ◽

Vol 24 (2) ◽

pp. 209-220 ◽

Cited By ~ 17

Author(s):

Alan Day

Keyword(s):

Abelian Groups ◽

Essential Extension ◽

Boolean Algebras ◽

Equational Class ◽

Injective Hull ◽

Fundamental Notion ◽

First Results ◽

Initial Results

The concept of injectivity in classes of algebras can be traced back to Baer's initial results for Abelian groups and modules in [1]. The first results in non-module types of algebras appeared when Halmos [14] described the injective Boolean algebras using Sikorski's lemma on extensions of Boolean homomorphisms [19]. In recent years, there have been several results (see references) describing the injective algebras in other particular equational classes of algebras.In [10], Eckmann and Schopf introduced the fundamental notion of essential extension and gave the basic relations that this concept had with injectivity in the equational class of all modules over a given ring. They developed the notion of an injective hull (or envelope) which provided every module with a minimal injective extension or equivalently, a maximal essential extension. In [6] and [9], it was noted that these relationships hold in any equational class with enough injectives.

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On Single Equational-Axiom Systems for Abelian Groups

Journal of the Australian Mathematical Society ◽

10.1017/s144678870000570x ◽

1969 ◽

Vol 9 (1-2) ◽

pp. 143-152 ◽

Cited By ~ 6

Author(s):

R Padmanabhan

Keyword(s):

Binary Operation ◽

Abelian Groups ◽

Unit Element ◽

Boolean Algebras ◽

Striking Result ◽

Nullary Operation ◽

Mathematical System ◽

Axiom Systems ◽

The Right

It is a fascinating problem in the axiomatics of any mathematical system to reduce the number of axioms, the number of variables used in each axiom, the length of the various identities, the number of concepts involved in the system etc. to a minimum. In other words, one is interested finding systems which are apparently ‘of different structures’ but which represent the same reality. Sheffer's stroke operation and. Byrne's brief formulations of Boolean algebras [1], Sholander's characterization of distributive lattices [7] and Sorkin's famous problem of characterizing lattices by means of two identities are all in the same spirit. In groups, when defined as usual, we demand a binary, unary and a nullary operation respectively, say, a, b →a·b; a→a−1; the existence of a unit element). However, as G. Rabinow first proved in [6], groups can be made as a subvariety of groupoids (mathematical systems with just one binary operation) with the operation * where a * b is the right division, ab−1. [8], M. Sholander proved the striking result that a mathematical system closed under a binary operation * and satisfying the identity S: x * ((x *z) * (y *z)) = y is an abelian group. Yet another identity, already known in the literature, characterizing abelian groups is HN: x * ((z * y) * (z * a;)) = y which is due to G. Higman and B. H. Neumann ([3], [4])*. As can be seen both the identities are of length six and both of them belong to the same ‘bracketting scheme’ or ‘bracket type’.

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More on entangled orders

Journal of Symbolic Logic ◽

10.2307/2695077 ◽

2000 ◽

Vol 65 (4) ◽

pp. 1823-1832

Author(s):

Ofer Shafir ◽

Saharon Shelah

Keyword(s):

Necessary Conditions ◽

Interesting Case ◽

Present Form ◽

Boolean Algebras ◽

Previous Version ◽

Simple Lemma ◽

New Construction ◽

Cardinal Arithmetic

This paper grew as a continuation of [Sh462] but in the present form it can serve as a motivation for it as well. We deal with the same notions, all defined in 1.1, and use just one simple lemma from there whose statement and proof we repeat as 2.1. Originally entangledness was introduced, in [BoSh210] for example, in order to get narrow boolean algebras and examples of the nonmultiplicativity of c.c-ness. These applications became marginal when other methods were found and successfully applied (especially Todorčevic walks) but after the pcf constructions which made their début in [Sh-g] and were continued in [Sh462] it seems that this notion gained independence.Generally we aim at characterizing the existence of strong and weak entangled orders in cardinal arithmetic terms. In [Sh462, §6] necessary conditions were shown for strong entangledness which in a previous version was erroneously proved to be equivalent to plain entangledness. In §1 we give a forcing counterexample to this equivalence and in §2 we get those results for entangledness (certainly the most interesting case). A new construction of an entangled order ends this section. In §3 we get weaker results for positively entangledness, especially when supplemented with the existence of a separating point (Definition 2.2). An antipodal case is defined in 3.10 and completely characterized in 3.11. Lastly we outline in 3.12 a forcing example showing that these two subcases of positive entangledness comprise no dichotomy. The work was done during the fall of 1994 and the winter of 1995. The second author proved Theorems 1.2, 2.14, the result that is mentioned in Remark 2.11 and what appears in this version as Theorem 2.10(a) with the further assumption den (I)θ < μ. The first author is responsible for waving off this assumption (actually by showing that it holds in the general case), for Theorems 2.12 and 2.13 in Section 2 and for the work which is presented in Section 3.

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The Ascending and Descending Varietal Chains of a Variety

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-004-2 ◽

1975 ◽

Vol 27 (1) ◽

pp. 25-31 ◽

Cited By ~ 7

Author(s):

B. Jónsson ◽

G. McNulty ◽

R. Quackenbush

Keyword(s):

Free Algebra ◽

Equational Class ◽

Similarity Type ◽

Nullary Operation ◽

Free Generators

Let F be a variety (equational class) of algebras. For n ≧ 0, Vn is the variety generated by the F-free algebra on n free generators while Vn is the variety of all algebras satisfying each identity of V which has no more than n variables. (Equivalently, Vn is the class of all algebras, , such that every n-generated subalgebra of is in V.) Note that unless nullary operation symbols are specified by the similarity type of V, V0 is the variety of all one element algebras while V° is the variety of all algebras.

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Value of new construction – major components: 1915–1964 [Current dollars]

Historical Statistics of the United States: Millennial Edition Online ◽

10.1017/isbn-9780511132971.dc1-255 ◽

2010 ◽

Author(s):

Kenneth A. Snowden

Keyword(s):

New Construction

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A New Construction of Incandescent Mantles

Scientific American ◽

10.1038/scientificamerican12201902-22549bsupp ◽

1902 ◽

Vol 54 (1407supp) ◽

pp. 22549-22549

Keyword(s):

New Construction

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Main Provisions Of The Concept Of Technology Of Diamond Cutting And Drilling Of Building Structures

Promyshlennoe i Grazhdanskoe Stroitel stvo ◽

10.33622/0869-7019.2019.06.59-63 ◽

2019 ◽

pp. 59-63

Keyword(s):

Combustion Engine ◽

Front Surface ◽

The Body ◽

Building Structures ◽

Diamond Cutting ◽

Variable Diameter ◽

Diamond Drilling ◽

Single Structure ◽

New Construction ◽

Non Destructive

The main provisions of the concept of technology of diamond cutting and drilling of building structures are considered. The innovativeness of the technology, its main possibilities and advantages are presented. Carrying out works with the help of this technology in underwater conditions expands its use when constructing and reconstructing hydraulic structure. The use of diamond drilling equipment with motors equipped with an internal combustion engine is considered. Drilling holes with a variable diameter during the reconstruction of the runways of airfields makes it possible to combine the landing mats into a single structure. The ability to cut inside the concrete mass, parallel to the front surface, has no analogues among the methods of concrete treatment. The use of this technology for producing blind openings in the body of concrete without weakening the structure is also unique. Work with precision quality in cutting and diamond drilling of concrete and reinforced concrete was noted by architects and began to be implemented in the manufacture of inter-room and inter-floor openings. Non-destructive approach to the fragmentation of building structures allows them to be reused. The technology of diamond cutting and drilling is located at the junction of new construction, repair, reconstruction of buildings and structures, and dismantling of structures. Attention is paid to the complexity and combinatorial application of diamond technology. Economic efficiency and ecological safety of diamond technology are presented. The main directions of further research for the development of technology are indicated.

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