A Note on Minors Determined by Clones of Semilattices

The C-minor partial orders determined by the clones generated by a semilattice operation (and possibly the constant operations corresponding to the identity or zero elements) satisfy the descending chain condition. 1. C-minors and C-decompositions Let A be a fixed nonempty base set. An operation on A is a map f : A → A for some integer n ≥ 1, called the arity of f . Denote by OA = ⋃ n≥1A A the set of all operations on A. The i-th n-ary projection (1 ≤ i ≤ n) is the operation (a1, . . . , an) 7→ ai, and it is denoted by x (n) i , or simply by xi when the arity is clear from the context. We say that the i-th variable is essential in f : A → A, if there exist elements a1, . . . , an, b ∈ A such that f(a1, . . . , ai−1, ai, ai+1, . . . , an) 6= f(a1, . . . , ai−1, b, ai+1, . . . , an). If the i-th variable is not essential in f , then we say that it is inessential in f . If f is an n-ary operation and g1, . . . , gn are m-ary operations, then the composition of f with g1, . . . , gn, denoted f(g1, . . . , gn) is the m-ary operation defined by f(g1, . . . , gn)(a) = f ( g1(a), . . . , gn(a) ) for all a ∈ A. A class of operations is a subset C ⊆ OA. A clone on A is a class C that contains all projections and is closed under composition. Let C be a class of operations on A. Let f and g be operations on A. We say that f is a C-minor of g, if f = g(h1, . . . , hm) for some h1, . . . , hm ∈ C, and we denote this fact by f ≤C g. We say that f and g are C-equivalent, denoted f ≡C g, if f and g are C-minors of each other. The C-minor relation ≤C is a preorder (i.e., a reflexive and transitive relation) on OA if and only if C is a clone. If C is a clone, then the C-equivalence relation ≡C is an equivalence relation on OA, and, as for preorders, ≤C induces a partial order 4C on the quotient OA/≡C. (See [1, 2].) Note that by the definition of C-minor, if C and K are clones such that C ⊆ K, then ≤ [C] ⊆ ≤ [K] and ≡ [C] ⊆ ≡ [K]. Let C be a clone on A. If f = g(φ1, . . . , φm) and φ1, . . . , φm ∈ C, we say that the (m + 1)-tuple (g, φ1, . . . , φm) is a C-decomposition of f : A n → A. We often avoid referring explicitly to the tuple and we simply say that f = g(φ1, . . . , φm) is a C-decomposition. Clearly, there always exists a C-decomposition of every f Date: September 18, 2008. 1