Semirigid Systems of Equivalence Relations

Abstract

A system M of equivalence relations on a set E is semirigid if only the identity and constant functions preserve all members of M. We construct semirigid systems of three equivalence relations. Our construction leads to the examples given by Zádori in 1983 and to many others and also extends to some infinite cardinalities. As a consequence, we show that on every set of at most continuum cardinality distinct from 2 and 4 there exists a semirigid system of three equivalence relations.

Hereditarily Rigid Relations: Dedicated to Professor I.G. Rosenberg on the Occasion of His 80-th Birthday

“…, a) / ∈ ρ, then choose b = a and define the partial constant function f b by dom (f b ) = {a} and f b (a) = b. Then f b ∈ pPol (1) ρ by definition. On the other hand, if (a, .…”

“…a projection or a constant function) (see, e.g. [1], [4], [6], [11]). We call these generalizations hereditarily rigid and hereditarily strongly rigid (rather than hereditarily semirigid and hereditarily strongly semirigid).…”

“…We call these generalizations hereditarily rigid and hereditarily strongly rigid (rather than hereditarily semirigid and hereditarily strongly semirigid). For ℓ ≥ 1, we define For simplicity, we have consider this condition instead pPol (1) ρ ⊆ Ω <ℓ (k) which would seem the natural generalization of semirigidity. We leave to a further study, the consideration of this generalization and we refer to Lemma 9 for a possible relationship between the two notions.…”

“…On the other hand, β n ℓ (X) identifies with the set of surjective maps from [n] onto X. Denote this number by s(n, ℓ) and recall that it satisfies the formula: (1) . (1). For (4) observe that a partial function f belongs to pPol (1) ρ if and only if every subfunction with domain of size at most h belongs to pPol (1) ρ.…”

“…Set pPol ρ := {f ∈ Par(k) | f preserves ρ} and Pol ρ := (pPol ρ) ∩ Op(k). Moreover let pPol (1) ρ := (pPol ρ)∩Par(k) (1) denote all partial unary functions that preserve ρ. Similarly let Pol (1) ρ := (Pol ρ) ∩ Op(k) (1) denotes all (total) unary functions that preserve ρ.…”