Results 221 to 230 of about 1,505,607 (258)
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Fuzzy Sets and Systems, 1996
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M. Atif, A. Mishref
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zbMATH Open Web Interface contents unavailable due to conflicting licenses.
M. Atif, A. Mishref
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Free Products and Free Products with Amalgamations
1982In this chapter, we shall describe the emergence of two new ideas which had a profound influence on the development of combinatorial group theory. Their creation is associated with three names: O. Schreier, who appeared prominently in Chapter II.3, E. Artin (1898–1962), and A. G. Kurosh (1908–1971).
Bruce Chandler, Wilhelm Magnus
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Products of Commutators in Free Products
International Journal of Algebra and Computation, 1997The genus of an element in the commutator subgroup of a group \(G\) is the minimal number of commutators of which the element is a product. It has been shown previously that in a free group each element of genus \(n\) can be obtained by permutation and suitable substitution on one of a finite number of words called orientable forms of genus \(n\).
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International Journal of Algebra and Computation, 1995
Topological methods are used to show that for certain subgroups S of a free product F, if w∈S is a commutator in F, then w is a commutator in S.
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Topological methods are used to show that for certain subgroups S of a free product F, if w∈S is a commutator in F, then w is a commutator in S.
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PRODUCTS OF COMMUTATORS IN FREE GROUPS
International Journal of Algebra and Computation, 2003Let F be the free group of rank 2, generated by x and y, and let w(x, y) ∈ F′ be a non-trivial word. We give elementary algebraic proofs and algorithms to (1) express [x, y]n as a product of [n/2] + 1 commutators and show this is the best possible; (2) show that (w(x, y))2 cannot be written as one w-word and if g ≠ 1 ∈ w(F) then show that the minimal ...
Mehri Akhavan-Malayeri, Akbar Rhemtulla
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1974
A comparative study which contrasts the theory of free groups with the theory of free products has some general interest. One area where comparison can sometimes be made in an elementary way concerns relations, and their consequences: for example, the consequences of the relation XY = YX .
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A comparative study which contrasts the theory of free groups with the theory of free products has some general interest. One area where comparison can sometimes be made in an elementary way concerns relations, and their consequences: for example, the consequences of the relation XY = YX .
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1993
Recall from Chapter 1 that a group is free if it has a presentation with no non-trivial relations. By Corollary 1.1.6 (c) every group is the homomorphic image of a free group and this indicates the central role that free groups play in combinatorial group theory.
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Recall from Chapter 1 that a group is free if it has a presentation with no non-trivial relations. By Corollary 1.1.6 (c) every group is the homomorphic image of a free group and this indicates the central role that free groups play in combinatorial group theory.
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Free Products of Pseudocomplemented Semilattices
Semigroup Forum, 2000Let \({\mathcal K}\) be the variety of pseudocomplemented semilattices (PCSs in brief), let \(S\), \(S_i\in{\mathcal K}\), for \(i\in I\). Then \(S\) is a free product of \(S_i\) \((i\in I)\) if there exist embeddings \(\varphi_i: S_i\to S\) such that \(S\) is generated by the set \(\bigcup \{\varphi_i(S_i);\;i\in I\}\), and if \(T\in K\) and \(\psi_i ...
Katriňák, T., Heleyová, Z.
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1995
The notion of generators and relations can be extended from abelian groups to arbitrary groups once we have a nonabelian analogue of free abelian groups. We use the property appearing in Theorem 10.11 as our starting point.
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The notion of generators and relations can be extended from abelian groups to arbitrary groups once we have a nonabelian analogue of free abelian groups. We use the property appearing in Theorem 10.11 as our starting point.
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FREE PRODUCTS OF NETWORKS AND FREE SYMMETRIZERS OF GRAPHS
Mathematics of the USSR-Sbornik, 1975Connected locally finite vertex-symmetric graphs, also called networks, are the subject of the article. The operation of a free product, preserving the vertex symmetry of graphs, is introduced. Properties of free products are studied, and a connection with free products of groups is established.
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