On Functions with a Unique Identification Minor

Abstract

We shed some new light to the problem of characterizing those functions of several arguments that have a unique identification minor. The 2set-transitive functions are known to have this property. We describe another class of functions that have a unique identification minor, namely functions determined by the order of first occurrence. We also present some examples of other kinds of functions with a unique identification minor. These examples have a relatively small arity.

Content and Singletons Bring Unique Identification Minors

“…It is well known that the 2-set-transitive functions have a unique identification minor (for a proof of this fact, see [13,Proposition 4.3]; this fact is also implicit in the work of Bouaziz, Couceiro, and Pouzet [3]). It was recently shown by the current author that the functions determined by the order of first occurrence also have this property [15,Corollary 5].…”

“…But then the relative order of the entries p and q is reversed in uσ, so f (uσ) = 0 1 = f (u), a contradiction. Proposition 3.7 [15,Proposition 7]. Assume that n > |A| + 1, and let f : A n → B.…”

“…As defined in [15], the map ofo : A * → A ♯ is given by the following rule: ofo maps each tuple a to the tuple obtained from a by deleting all repeated occurrences of symbols, retaining only the first occurrence of each symbol; in other words, ofo(a) lists the symbols occurring in a in the order of first occurrence (hence the initialism ofo). We also say that a function determined by ofo is determined by the order of first occurrence.…”

The Minor Order of Homomorphisms via Natural Dualities

“…We retrieve the following result as a consequence of Proposition 4.2. Functions f ∈ F AB that have a unique identification minor have received special interest and have been put in relation with their invariance group (see [20]). Recall that the invariance group Inv(f…”

“…The general idea behind these investigations is to determine how much information about an operation f can be retrieved from its minors. In particular, a series of recent papers deals with reconstruction properties, asking which operations f can be recovered from (a portion of) their minor posets (see [18,19,20,21]).…”