A natural extension of the Wadge hierarchy is obtained if we consider k-partitions (or even Q-partitions for a countable better quasiorder Q) of the Baire space instead of sets (i.e., 2-partitions). Natural initial segments of the Q-Wadge hierarchy were recently characterised in terms of the so called h-quasiorder on suitably iterated Q-labeled countable well founded forests. Though these characterisations are natural and constructive, their definitions are rather long and technical. In this paper, we find clear shorter characterizations as (reducts of) a kind of free structures satisfying some natural axioms. Informally, we provide short and clear axiomatisations of the initial segments.
B.A.Davey and H.A.Priestley, Introduction to Lattices and Order, Cambridge, 1994.
3.
P.Hertling, Topologische Komplexitätsgrade von Funktionen mit endlichem Bild, in: Informatik-Berichte, Vol. 152, Fernuniversität Hagen, 1993.
4.
A.S.Kechris, Classical Descriptive Set Theory, Springer, New York, 1995.
5.
T.Kihara and A.Montalban, On the structure of the Wadge degrees of BQO-valued Borel functions, Trans. Amer. Math. Soc.371(11) (2019), 7885–7923. doi:10.1090/tran/7621.
6.
T.Kihara and V.Selivanov, Wadge-like degrees of Borel bqo-valued functions, 2019, arXiv:1909.10835v1.
7.
K.Kuratowski and A.Mostowski, Set Theory, North Holland, 1967.
8.
C.St.J.A.Nash-Williams, On well-quasiordering infinite trees, Proc. Cambridge Philos. Soc.61 (1965), 697–720. doi:10.1017/S0305004100039062.
9.
V.Selivanov, Q-Wadge hierarchy in quasi-Polish spaces, 2019, arXiv:1911.02758v1.
10.
V.L.Selivanov, Boolean hierarchies of partitions over reducible base, Algebra and Logic43(1) (2004), 44–61. doi:10.1023/B:ALLO.0000015130.31054.b3.
11.
V.L.Selivanov, The quotient algebra of labeled forests modulo h-equivalence, Algebra and Logic46(2) (2007), 120–133. doi:10.1007/s10469-007-0011-5.
V.L.Selivanov, Fine hierarchies and m-reducibilities in theoretical computer science, Theoretical Computer Science405 (2008), 116–163. doi:10.1016/j.tcs.2008.06.031.
14.
V.L.Selivanov, Towards a descriptive theory of cb0-spaces, Mathematical Structures in Computer Science.28(8) (2017), 1553–1580. doi:10.1017/S0960129516000177.
15.
V.L.Selivanov, Well quasiorders and hierarchy theory, in: Well Quasiorders in Computation, Logic, Language and Reasoning, P.M.Schuster, M.Seisenberger and A.Weiermann, eds, Series Trends in Logic, Springer, 2020, pp. 271–320, arXiv:1809.02941. doi:10.1007/978-3-030-30229-0_10.
16.
V.L.Selivanov, Extending Wadge theory to k-partitions, in: CiE 2017, J.Kari, F.Manea and I.Petre, eds, LNCS, Vol. 10307, Springer, Berlin, pp. 387–399.
17.
S.G.Simpson, Bqo-theory and Fraïssé conjecture, in: Recursive Aspects of Descriptive Set Theory, R.Mansfield, G.Weitkamp, eds, Oxford University Press, New York, 1985, Chapter 9.
18.
F.van Engelen, A.Miller and J.Steel, Rigid Borel sets and better quasiorder theory, Contemporary mathematics65 (1987), 199–222. doi:10.1090/conm/065/891249.
19.
R.Van Wesep, Wadge degrees and descriptive set theory, Lecture Notes in Mathematics689 (1976), 151–170.
20.
W.W.Wadge, Reducibility and determinateness in the Baire space, PhD thesis, University of California, Berkely, 1984.
