## Linearization of definable order relations.(English)Zbl 0942.03055

It is well known that every partial quasi-order (PQO) is linearizable, i.e., it can be extended to a linear order on the same domain. However not every Borel PQO is Borel linearizable. For example, if $$E_0$$ is an equivalence relation defined on $$2^{\omega}$$ as follows: $$\langle a,b\rangle \in E_0$$ iff $$a(k)=b(k)$$ for all but finite $$k$$, then it is not Borel linearizable [see L. A. Harrington, D. Marker and S. Shelah, Trans. Am. Math. Soc. 310, 293-302 (1988; Zbl 0707.03042)]. Similarly, the anti-lexicographical partial order $$\leq_0$$ on $$2^{\omega}$$ is not Borel linearizable. In fact, the author proved that $$\leq_0$$ is a minimal Borel-nonlinearizable Borel order in the following sense: for every Borel PQO $$\preceq$$ on $${\mathcal N}=\omega^{\omega}$$ either: (i) $$\preceq$$ is Borel linearizable, or (ii) there exists a continuous half-order-preserving 1-1 map $$F\colon \langle 2^{\omega}, \leq_0\rangle\to \langle{\mathcal N}, \preceq\rangle$$ such that $$\langle a,b\rangle \not\in E_0$$ implies $$F(a)\not\preceq F(b)$$ [see V. Kanovei, Fundam. Math 155, 301-309 (1998; Zbl 0909.03041)]. The paper contains similar dichotomical linearization theorems for some non-Borel partial orders, including:
$$\bullet$$ analytic and bi-$$\kappa$$-Souslin ($$\kappa>\omega_1$$) orders [see S. Shelah, Isr. J. Math 47, 139-153 (1984; Zbl 0561.03026)];
$$\bullet$$ definable orders in the Solovay model [R. M. Solovay, Ann. Math., II. Ser. 92, 1-56 (1970; Zbl 0207.00905)].

### MSC:

 03E15 Descriptive set theory 03E35 Consistency and independence results 06A05 Total orders 06A06 Partial orders, general

### Citations:

Zbl 0707.03042; Zbl 0909.03041; Zbl 0561.03026; Zbl 0207.00905
Full Text:

### References:

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