By Thomas Keilen

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Wr ) where each wi is an Ω -word. 2) wi ∈ X ∪ Ω (0) and v(wi ) = 1, or wi = (ω , a1 , . . , an ) where n ≥ 1, ω ∈ Ω (n) and a j ∈ FΩ (X). Then v(a j ) = 1 and each tail of a j has positive valence for 1 ≤ j ≤ n since (a j ) < (w), and hence v(wi ) = 1 − n + n = 1 and v(w) = r. Also, if u is a proper tail of w, then u = (t, wi+1 , . . , wr ) where either t is a proper tail of wi , or t = wi and i ≥ 2. In the second case v(u) = r − i + 1 ≥ 1. 4 Universal Algebra 25 the first case t = (s, a j+1 , .

An ). More explicitly, if ai = (ai1 , . . , aimi ) for 1 ≤ i ≤ n, where ai j ∈ Ω ∪ X, then ω (a1 , . . , an ) = (ω , a11 , . . , a1m1 , a21 , . . , anmn ). An element of the algebra W (Ω , X) will be called an Ω -row in X. The subalgebra of W (Ω , X) generated by X, which is denoted by FΩ (X) or F(X), is called the Ω -word algebra on X, and its elements will be called Ω -words in the alphabet X. 2) the elements of FΩ (X) are either in X0 = X ∪ Ω (0) or are in Xk for some k ≥ 1. The elements in X1 \X0 are of the form ω (a1 , .

Show that each fully invariant congruence S on FΩ (X) is the set of identities of some variety of Ω -algebras; that is, S = S∗∗ . ) 30 1 Partially Ordered Sets and Lattices 16. Let V be the variety of Ω -algebras generated by the class C ⊆ A (Ω ). Show that the algebra A is in V iff A is a homomorphic image of a subalgebra of a direct product of a family of algebras in C . 17. Show that an Ω -algebra A is in a variety V iff each finitely generated subalgebra of A is in V . 18. An algebra A is called a generic algebra in the variety V if V is the variety generated by {A}.