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-Ary Free Lattices from Finitary Ones
Let be an infinite regular cardinal. A poset L is called an -lattice if and only if for all X L satisfying 0 < |X| < m, ∧ X and ∨ X exist.
This paper is a part of a sequence of papers, [5], [6], [7], [8], developing the theory of -lattices. For a survey of some of these results, see [9].
The -lattice 
is described in [6]; γ denotes the zero and γ′ the unit of . In particular, formulas for -joins and meets are given. (We repeat the essentials of this description in Section 4.)
In [6] we proved the theorem stated below. Our proof was based on characterization of 
(the free -lattice on P) due to [1]; as a result, our proof was very computational.
It is well known that given the polynomial algebra 
(for definitions, see §2), an algebra 
of type τ, and a sequence a of elements of 
, one can define a congruence relation θa of 
such that the factor algebra 
is isomorphic to the subalgebra of 
generated by a, and the isomorphism is given in a very simple way.
