# Equivalence relations: Cosimplicial semilattice?

For $$n\ge 0$$, let $$E_n$$ be the set of all equivalence relations on $$[n]:=\{0,\dotsc,n\}$$. Now given two equivalence relations $$R,R'\in E_n$$, we build their join $$R\vee R' := \langle R\cup R'\rangle,$$ the smallest equivalence relation containing $$R$$ and $$R'$$. The following seems to be clear:

1. $$(E_n,\vee)$$ is a bounded join-semilattice, i. e. it is a commutative monoid (with $$1=\Delta$$, the smallest equivalence relation) where each element is idempotent.

2. Consider the cosimplicial maps $$d_i:[n]\to [n+1]$$ and $$s_i:[n]\to [n-1]$$ and define maps $$D_i:E_n\to E_{n+1}$$ and $$S_i:E_n\to E_{n-1}$$ by $$D_iR:=\langle d_i^{\times 2}R\rangle~~~\text{ and }~~~S_iR:=\langle s_i^{\times 2}R\rangle$$ Then they should satisfy the cosimplicial identities and moreover, $$D_i(R\vee R')=D_iR\vee D_iR'$$ and $$S_i(R\vee R')=S_iR\vee S_iR'$$.

All in all, this should give us a cosimplicial object in the category of bounded semilattices.

If I didn’t make any obvious mistake, this looks to me as a construction which must be well-known and written down somewhere, maybe in a different terminology. Thus, I assume that I missed something which does not work so easily. Does someone find a mistake or know the general construction from the literature?