Geodesic
Joins and Direct Products of Equational Classes
Canadian mathematical bulletin, Tome 12 (1969) no. 6, pp. 741-744

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Grätzer, G.; Lakser, H.; Płonka, J. Joins and Direct Products of Equational Classes. Canadian mathematical bulletin, Tome 12 (1969) no. 6, pp. 741-744. doi: 10.4153/CMB-1969-095-x
@article{10_4153_CMB_1969_095_x,
     author = {Gr\"atzer, G. and Lakser, H. and P{\l}onka, J.},
     title = {Joins and {Direct} {Products} of {Equational} {Classes}},
     journal = {Canadian mathematical bulletin},
     pages = {741--744},
     year = {1969},
     volume = {12},
     number = {6},
     doi = {10.4153/CMB-1969-095-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-095-x/}
}
TY  - JOUR
AU  - Grätzer, G.
AU  - Lakser, H.
AU  - Płonka, J.
TI  - Joins and Direct Products of Equational Classes
JO  - Canadian mathematical bulletin
PY  - 1969
SP  - 741
EP  - 744
VL  - 12
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-095-x/
DO  - 10.4153/CMB-1969-095-x
ID  - 10_4153_CMB_1969_095_x
ER  - 
%0 Journal Article
%A Grätzer, G.
%A Lakser, H.
%A Płonka, J.
%T Joins and Direct Products of Equational Classes
%J Canadian mathematical bulletin
%D 1969
%P 741-744
%V 12
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-095-x/
%R 10.4153/CMB-1969-095-x
%F 10_4153_CMB_1969_095_x

[1] 1. Astromoff, A., Some structure theorems for primal and categorical algebras. Math. Z. 87 (1965) 365–377. Google Scholar

[2] 2. Chang, C. C., Jónsson, B., and Tarski, A., Refinement properties for relational structures. Fund. Math. 55 (1964) 249–281. Google Scholar

[3] 3. Day, A., A characterization of modularity for congruence lattices of algebras. Canad. Math. Bull. 12 (1969) 167–173. Google Scholar

[4] 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] 5. Grätzer, G., Universal algebra. (The University Series in Higher Mathematics, D. Van Nostrand Co. Inc., Princeton, N.J., 1968). Google Scholar

[6] 6. Grätzer, G., Mal' cev type conditions. J. Comb.Theory (to appear). Google Scholar

[7] 7. Kelenson, P., Thesis. (Berkeley, 1969). Google Scholar

[8] 8. Kimura, N., Note on idempotent semigroups, III. Proc. Japan Acad. 34 (1958) 113–114. Google Scholar

[9] 9. Kimura, N., The structure of idempotent semigroups, I. Pacific J. Math. 8 (1958) 257–275. Google Scholar

Cité par Sources :