On essential variables of dual operations and the consequences for operations (Q364683)

Let \(C\) be a category, and let \(X\), \(Y\) be objects in \(C\). Assume that for every \(n\in{\mathbb N}\), the \(n\)th copower \(n\cdot X\) exists in \(C\), and for \(i \leq n\) let \(\iota_i^n: X\to n\cdot X\) denote the associated injection morphism. An \(n\)-ary dual operation (co-operation) is a morphism \(g: Y\to n\cdot X\). Dualizing the concept of essential variables for operations, the author calls the \(i\)th variable of \(g\) nonessential if \[ [\iota_1^{n+1},\dots,\iota_n^{n+1}]\circ g = [\iota_1^{n+1},\dots,\iota_{i-1}^{n+1},\iota_{n+1}^{n+1},\iota_{i+1}^{n+1},\dots,\iota_n^{n+1}]\circ g. \] Here \([\iota_1^{n+1},\dots,\iota_n^{n+1}]: n\cdot X \to (n+1)\cdot X\) denotes the cotupling of the injections. The essential arity of \(g\) is the number of essential (that is, not nonessential) variables. Contrary to intuition, essentially nullary dual operations are not necessarily the same as constant dual operations. This paper is the start of a general investigation of essential variables, essential arities and the arity gap of dual operations. The results are applied to three particular categories: Boolean groups (groups of exponent \(2\)), bounded posets, and bounded strongly complemented posets. Via duality theory, information on dual operations is then transferred to operations for the corresponding dual categories, namely, Boolean groups, distributive lattices, and median algebras.