Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-16T15:42:17.087Z Has data issue: false hasContentIssue false
  • English
  • Français

Relations between the amalgamation property and algebraic equations

Published online by Cambridge University Press:  09 April 2009

Harald Hule
Harald Hule
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brazil.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A variety B is called solutionally complete if any system of algebraic equations over an algebra A in B has a solution in A provided it is solvable in B and has at most one solution in any extension of A in B. B is called solutionally compatible if every solvable system of equations over an algebra in B is also solvable over any extension of that algebra. It is shown that solutional compatibility is equivalent with the amalgamation property and that a weaker form of the strong amalgamation property is sufficient but not necessary for equational completeness.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 3 , May 1978 , pp. 257 - 263
Copyright
Copyright © Australian Mathematical Society 1978

References

Grätzer, G. (1971), Lattice Theory (Freeman, San Francisco).Google Scholar
Hule, H. (1976), ‘Über die Eindeutigkeit der Lösungen algebraischer Gleichungssysteme’, J. reine angew. Math. 282, 157–161.Google Scholar
Hule, H. and Müller, W. B. (1976), ‘On the compatibility of algebraic equations with extensions’, J. Austral. Math. Soc. 21 (Series A), 381–383.CrossRefGoogle Scholar
Ježek, J. (1976), ‘EDZ-varieties: the Schreier property and epimorphisms onto’, Comment. Math. Univ. Carolinae 17, 281–290.Google Scholar
Jónsson, B. (1961), ‘Sublattices of a free lattice’, Canad. J. Math. 13, 256–264.CrossRefGoogle Scholar
Lausch, H. and Nöbauer, W. (1973), Algebra of Polynomials (North-Holland, Amsterdam).Google Scholar