Sound, totally sound, and unsound recursive equivalence types

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Abstract

An independent subset I of V is called sound is I is contained in a r.e. independent set. If not, we shall call I unsound. We call an RET A totally sound if every independent set in A is sound. Clearly every RET containing an r.e. set is totally sound, and it had been suggested that the converse held: viz, every totally sound RET contained a recursive set.

However, we show that there are 2χ0 sets {Aπ} such that if BmAπ then RET(B) is totally sound. In the co-r.e. case it is shown that if A is co-r.e. nonrecursive and nonisolic, then RET(A) is not totally sound. Indeed, RET(A) contains an independent set which is a basis of a subspace (of codimension 1 in V), no basis of which is sound. This result is deduced from a construction of a new type of r.e. subspace of V. That is, we show that if R is a fully co-r.e. nonrecursive subspace of V there existse an r.e. subspace V sich that VR = V and V has the property that if W is an r.e. subspace with WV, then dim(V/W) < ∞ implies W = V.

On the other hand , the co-simple isols provide a nice dichotomy. First, it is shown that (in each high r.e. degree) there is a co-simple totally sound RET. Second it is shown that if a is any nonzero r.e. degree, there exists a nonzero r.e. degree ⩽Ta such that b bounds no nontrivial totally sound co-r.e. RET. That is, if B is r.e. with Turing degree b and C is co-r.e. with CTB, then either (i) C is recursive or (ii) RET(C) is not totally sound.

Finally, we extend our results to general algebraic settings; first to a class of Steinitz closure systems, and later to a general model-theoretic setting (without dependence).

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Partially supported by Grant (NUS) RP-85/83 whilst the author held a position at the National University of Singapore.