# Errors in Relation Algebras by Games

We would be happy to hear of any further errors and we will list them here giving due credit.

1. Page 28 line 13: typo: there should be no comma after \cal B.

2. Page 42 line 21: this is wrong.  In ZF, the boolean prime ideal theorem (BPI) implies that any set can be linearly ordered, but not conversely: see the habilitation thesis of Ulrich Felgner.  This most unfortunate mistake is ours; [Hod93] does not make this mistake, and it is doubly unfortunate that the text rather reads as though it does.  We apologise to Wilfrid Hodges for this.  (We spotted this mistake in about 2003.  Thanks to Richard Kramer for also noticing it more recently.)

3. Page 86 line 11: replace the comma before "we" at the end of the line by a colon.

4. Page 94, theorem 2.94: the 'proof' given of this theorem is desultory and does not give necessary credit to the paper [GivVen99] (S. Givant and Y. Venema, The preservation of Sahlqvist equations in completions of Boolean algebras with operators, Algebra Universalis 41 (1999) 47-84).  We apologise for this.  Here is a sketch proof.  For a Sahlqvist equation E with correspondent C, and a conjugated atomic BAO A (of the signature of E), the following are equivalent:
1. E holds in A
2. E holds in the completion of A (by [GivVen99, corollary 34(ii)], using that A is conjugated)
3. E holds in the full complex algebra over At(A) (since this is the completion of A, up to isomorphism)
4. C holds in At(A) (by [RijVen95, theorem 3.3]; this paper also discusses the construction of C by an algorithm)
If A is of the form Cm(S), it is already complete, the equivalence of 1 and 2 is not needed, and the conjugated requirement can be dropped.  If E is (in the terminology of [GivVen99]) a simple Sahlqvist equation, the 'conjugated' requirement can be weakened to completely additive in the equivalence of 1 and 2, by using [GivVen99, corollary 34(i)] instead.
We (again) thank Steven Givant for pointing out this error (March 2015).

5. Further, just below on page 94, the proof of theorem 2.96 mentions as one possibility to 'use theorem 2.94 and remark 2.67'.  This is circular and should be deleted, since theorem 2.96 is just [GivVen99, corollary 34(ii)], as used in the equivalence of 1 and 2 above (part of the proof of 2.94).

6. Page 118 Problem 3.27: we misquoted this, and rather nonsensically.  The original problem P2 from [Madd94a] (attributed to Jónsson) is "Find all simple relation algebras with no nontrivial subalgebras", and a relation algebra is said to have no nontrivial subalgebras if "the only subalgebras...are itself and the minimal subalgebra with elements 0, 1, 1', and 0' ".
Thanks to Steven Givant for pointing this out.

7. Page 168 line -4: we should have mentioned that RCAα was proved to be a variety in [Tar55].

8. Page 173 lines 4,5: delete "of w (or equivalently of w')".

9. Page 186 line -8: cja should be cia.  (Thanks to Agi Kurucz for spotting this, and to Daniel Rogozin for also noticing it subsequently.)

10. Page 209: in the definition of a 'point' in item 3, the inequality a;1;a≤1' should be ã;1;a≤1', where we have used ã to denote the converse of a.

11. Page 394, exercise 4, line 8: 12.2(11) should be 12.3(11).

12. Page 519 lines 8,9,15-18: all R and R' here should have the subscript _S.  (Thanks again to Agi Kurucz for this.)

13. Page 520: as may well be clear from remark 17.12, in the context of problem 17.13 we are tacitly adopting a wider definition of 'universal formula' than the (prenex) one given on p.33 — namely, a first-order formula in which every ∃ lies under an odd number of negations. (We take the primitive operations of first-order formulas to be ∧, ¬, and ∃ — see p.31.)

14. Page 546 line -2: this should read "if RA(τ) is representable then τ is a yes-instance of the tiling problem".

15. Page 560: at the end of section 18.7.5 we should have remarked that T1 clearly holds. No, this is an error in the errata: T1 and T2 for this case (and other cases) are checked later, on p.567.

16. Page 603 line 9: insert 'finite algebra on finite base property' before 'fails'.

17. Page 627, item 13 (problem 5.47) of problem list: the part asking whether RaCAn is closed under subalgebras for finite n≥4 should be deleted, since it is already answered negatively in fact 5.46(4) on p.192.  We thank Hajnal Andréka and István Németi for pointing this out.  The version of problem 5.47 on p.192 does not have this part.  (Some other problems, such as 12.38 and 18.17, are also stated slightly differently in the main text and in the 'problems' chapter 21.)

18. Page 629: at the start of the bibliography we state that "Numbers in brackets at the end of each reference refer to the pages on which it is cited". These numbers were compiled by a rewritten latex \cite command, which seems to have given some inaccurate results.  Where the citation occurs near the end of a page, the page number is sometimes that of the following page --- one too high.  For example, [SteVen98] is listed as being cited on page 388, among others, whereas the citation is actually at the end of page 387.  [She71] is another example.

19. Bibliography: J. Monk should be J. D. Monk throughout.  Our apologies.

20. Page 644: the page numbers of [Ly61] should be 21-28.

21. Page 648: the page numbers of [Mon91] should be 723-726.

22. Page 678: in the index entry "hyperbasis:restriction of", page 372 should be 373.

23. Page 687: in the index entry for "simple equation", add page 183.

24. Page 690: in the index entry for "undecidability:modal logics", add page 541.
Written October 2010 with minor rewording January 2011; latest additions Sep 2022.