Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-04-30T13:52:15.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Joins and Direct Products of Equational Classes

Published online by Cambridge University Press:  20 November 2018

G. Grätzer ,
H. Lakser  and
J. Płonka
G. Grätzer
Affiliation:
University of Manitoba
H. Lakser
Affiliation:
University of Manitoba
J. Płonka
Affiliation:
University of Manitoba
Rights & Permissions [Opens in a new window]

Extract

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.

Let K0 and K1 be equational classes of algebras of the same type. The smallest equational class K containing K0 and K1 is the join of K0 and K1; in notation, K = K0 ∨ K1. The direct product K0 × K1 is the class of all algebras α which are isomorphic to an algebra of the form a0 × a1, a0 ∈ K1. Naturally, K0 × K1 ⊆ K0 ∨ K1, Our first theorem states a very simple condition under which K0 × K1 = K0 ∨ K1, and an additional condition under which the representation α ∨ a0 × a1 unique.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 6 , December 1969 , pp. 741 - 744
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Astromoff, A., Some structure theorems for primal and categorical algebras. Math. Z. 87 (1965) 365–377.Google Scholar
2. Chang, C. C., Jónsson, B., and Tarski, A., Refinement properties for relational structures. Fund. Math. 55 (1964) 249–281.Google Scholar
3. Day, A., A characterization of modularity for congruence lattices of algebras. Canad. Math. Bull. 12 (1969) 167–173.Google Scholar
4. Foster, A. L., The identities of - and unique subdirect factorization within - classes of universal algebras. Math. Z. 62 (1955) 171–188.Google Scholar
5. Grätzer, G., Universal algebra. (The University Series in Higher Mathematics, D. Van Nostrand Co. Inc., Princeton, N.J., 1968).Google Scholar
6. Grätzer, G., Mal' cev type conditions. J. Comb.Theory (to appear).Google Scholar
7. Kelenson, P., Thesis. (Berkeley, 1969).Google Scholar
8. Kimura, N., Note on idempotent semigroups, III. Proc. Japan Acad. 34 (1958) 113–114.Google Scholar
9. Kimura, N., The structure of idempotent semigroups, I. Pacific J. Math. 8 (1958) 257–275.Google Scholar