The associative law

Algebra ◽  
2004 ◽  
pp. 11-12
Author(s):  
Israel M. Gelfand ◽  
Alexander Shen
Keyword(s):  
Associative Law

On semi-groups

1940 ◽  
Vol 36 (4) ◽  
pp. 387-400 ◽  
Author(s):  
D. Rees
Keyword(s):  
General Problem ◽  
Associative Law

This paper is an attempt to apply the methods of the theory of algebras to the more general problem of the structure of semi-groups, i.e. systems with one composition, satisfying the associative law.

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

The Associative Law

1958 ◽  
pp. 23-55
Author(s):  
Richard Hubert Bruck
Keyword(s):  
Associative Law

The Ring Scheme Fg

1972 ◽  
Vol 15 (1) ◽  
pp. 79-85 ◽  
Author(s):  
Ian G. Connell
Keyword(s):  
Formal Group ◽  
Associative Law ◽  
Image Position ◽  
Surprising Observation

Let A be a ring and s2(A) the set of idempotents in A. It is a familiar fact that s2(A) becomes a ring if we define 0, 1 and multiplication as in A, and a new negative and new addition by1R. A. Melter [3] made the surprising observation that s3(A), where in general sq(A)={x∈A:xq=x}, is a ring if 2 is a unit in A and we define 0,1 minus and multiplication as in A, and a new addition by2The nonobvious facts are that s3(A) is closed under ⊕ and that ⊕ is associative when applied to the elements of s3(A). The ⊕ in (1) is actually a formal group over A, and so is associative when applied to any elements of A. However in (2) (and similarly in other cases we shall define) the ⊕ is not a formal group, and the associative law depends on the fact that the elements involved are in s3(A).

Partial clones

2020 ◽  
Vol 13 (08) ◽  
pp. 2050161
Author(s):  
Klaus Denecke
Keyword(s):  
Computer Science ◽  
Natural Numbers ◽  
Tree Languages ◽  
Associative Law ◽  
The Many ◽  
Composition Of Operations

A set [Formula: see text] of operations defined on a nonempty set [Formula: see text] is said to be a clone if [Formula: see text] is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the [Formula: see text]-ary operations defined on set [Formula: see text] for all natural numbers [Formula: see text] and the operations are the so-called superposition operations [Formula: see text] for natural numbers [Formula: see text] and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set [Formula: see text] and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.

The generalized associative law in vague groups and its applications—II

2006 ◽  
Vol 176 (11) ◽  
pp. 1488-1530 ◽  
Author(s):  
M DEMIRCI
Keyword(s):  
Associative Law

The generalized associative law in smooth groups

2005 ◽  
Vol 169 (3-4) ◽  
pp. 227-244
Author(s):  
M DEMIRCI
Keyword(s):  
Associative Law
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