arXiv subjects
Publications and source records attributed to Bea Adam-Day.
We answer some questions about graphs which are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.
It is known that, if we take a countable model of Zermelo--Fraenkel set theory ZFC and "undirect" the membership relation (that is, make a graph by joining $x$ to $y$ if either $x\in y$ or $y\in x$), we obtain the Erdős--Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation, so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel's Anti-Foundation Axiom). The resulting graph may fail to be simple, it may have loops (if $x\in x$ for some $x$) or multiple edges (if $x\in y$ and $y\in x$ for some $x,y$). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the "random loopy graph" (which is $\aleph_0$-categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $\aleph_0$-categorical, but has infinitely many $1$-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.