Is there any name in the literature of functions satisfying this order property? The property is that for $f:D\to D$ is a function on some partially order set: $f(x)\geq x$ for every $x\in D$. That is $f≥I$ where $I$ is the identity function on $D$. Of course we can replace $\geq $ by $\leq$. In some settings we know that projections ($p^2=p$) satisfy the property for $\leq$.
Thanks in advance.
 A: The property $(\forall x)\,x\leq f(x)$ was called "inflationary" in Peter Freyd's paper "Aspects of topoi". Years later, Yuri Gurevich was working on a paper in theoretical computer science and asked me what I called this property.  I told him "inflationary" but mentioned that this was not, as far as I knew, standard terminology.  Yuri liked the name and used it, and it seems to have caught on among people working in computer science and finite model theory, so that, for example, "inflationary fixed point" is now standard terminology.
A: There is a related usage in set theory in the context of Fodor's lemma, where we often consider functions $f$ on the ordinals with the property that $f(\alpha)<\alpha$. These are called the regressive functions, and sometimes they are called pressing-down functions; I suppose we could also now call them strictly deflationary. Fodor's lemma asserts that every regressive function on an uncountable regular cardinal is constant on a stationary set. 
A: I think if we use the term "progressing" it will be reasonable to use "regressing" for the contrary. I have found the term progressing in the following reference:
Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed., 2010).
But, I would be gratful if one can provide me some references in which there is a use of the term "inflationary".
Many thanks everyone.
A: The property $∀x:x≤f(x)$ is also called „extensive“, especially with respect to closure systems or lattices.
In general, a closure system is defined as the image of a closure operator, which is an idempotent and extensional endomorphism of an ordered set.
