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

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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
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     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/}
}
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