A marriage of groups and Boolean algebras. In memory of Steven R. Givant

1. A marriage of groups and Boolean algebras.In memory of Steven R. Givant Andréka, H. and Németi, I. Rényi Institute of Mathematics seminar, December 21, 2018.

2. Relation Algebras Already in his 1941 article, Tarski remarked that the theory of relation algebras seemed to be a kind of union of the theories of Boolean algebras and of groups. The penultimate theorem we shall discuss provides an explanation of this connection. RA = Groups + BA

3. GROUPS Group: (A, •, 1’, -1) Brandt groupoid: (A, •, I,-1) Polygroupoid: (A, •, I,-1) • binary operation • partialbinary operation • many-valuedbinary operation a•b in A, for all a,b a•b in A, for somea,b a•bsubset of A, for all a,b Invertible monoid Invertible monoid Invertible monoid

4. Invertible monoid Invertible monoid Group: (A, •, 1’, -1) Polygroupoid: (A, •, I,-1) Brandt groupoid: (A, •, I,-1) • binary operation • partialbinary operation • multivaluedbinary operation a•b subset of A, for all a,b a•b in A, for all a,b a•b in A, for some a,b • associative: a(bc)=(ab)c • associative: a(bc)=(ab)c if ab, bc exist • associative: a(bc)=(ab)c Invertible monoid 1’ is identity: a1’ = 1’a =a I is set of identities: aI=Ia=a e in I is identity: ae= a, ea =a, if exist -1is inverse: aa -1 = a -1 a =1’ -1is inverse: multivalued version -1is inverse: aa -1a=a, and aa -1a exists

5. Invertible monoid Polygroupoid: (A, •, I,-1) • multivaluedbinary operation a•b subset of A, for all a,b • associative: a(bc)=(ab)c complex multiplication b a I is identity: aI=Ia=a c -1is inverse: a in bciff b in ac -1iff c in b -1a . Complex multiplication: XY = unionof { ab : a in X, b in Y}

6. GROUPS

7. Cayley representation Cayley representation: A is set of permutations on a set Composition, identity map, inverse 3 Group: (A, •, 1’, -1) 1 2 • binary operation 2 1 0 Suc = { (0,1), (1,2), (2,3), (3,0) } 0 3 0 1 2 3

8. Inverse Cayley representation: A is set of permutations on a set Composition, identity map, inverse 3 Group: (A, •, 1’, -1) 1 2 • binary operation 2 1 0 Suc = { (0,1), (1,2), (2,3), (3,0) } 0 3 0 1 2 3 Suc-1 = { (1,0), (2,1), (3,2), (0,3) }

9. Composition, identity Cayley representation: A is set of permutations on a set Composition, identity map, inverse 3 Group: (A, •, 1’, -1) 1 2 • binary operation 2 1 0 Suc = { (0,1), (1,2), (2,3), (3,0) } 0 3 0 1 2 3 Suc-1 = { (1,0), (2,1), (3,2), (0,3) } Suc2 = { (0,2), (1,3), (2,0), (3,1) }

10. Cayley representation of Z6

11. BRANDT GROUPOIDS

12. Brandt groupoid structure Structure: Copies of a group on the full graph on I p q r G p q r G G q Brandt groupoid: (A, •, I, -1) G G • partial binary operation r p A = { (i,g,j) : i,j in I and g in G } (i,g,j)•(j,h,k) = (i, gh, k) multiplication in G G

13. Brandt groupoid with I={p,q} and G=Z3 G G G Category

14. POLYGROUPOIDS

15. Polygroupoid structure Structure: Theorem (Comer, 1983) Polygroupoids are exactly atom-structures of atomic relation algebras. RA = SCm PG. Polygroupoid: (A, •, I, -1) Representation of a polygroupoid: Elements of A with binary relations • as composition of binary relations I as identity relation -1 as converse of a relation • many-valued binary operation Complete representations of RA: determined by polygroupoid Incomplete representations of RA: determined by BA structure Subject of second part of the talk

16. STORY • Representation theorem of Jónsson and Tarski 1952 • Discovery of Roger Maddux 1991 • Idea of Steven Givant 1991 • Vision of Steve Comer 1983

17. Loop polygroupoids a is a loop if there is x in I such that xax=a. A polygroupoid is a loop-polygroupoidiff the product on loops is a partial function. LPG A relation algebra is measurable iff the identity is the sum of atoms, and for each subidentity atom x the square x;1;x is the supremum of functional elements. MRA The structure of LPGs is very similar to BGs: Groups on the vertices, but different groups possible, Factor groups on the edges. Plus a common factor group in the middle of each triangle.

18. LOOP POLYGROUPOIDS Gijk Gij G Gik G Gjk G Gi G G Gj A = { (i,g,j) : i,j in I and g in Gij } A = { (i,g,j) : i,j in I and g in G } (i,g,j)•(j,h,k) = { (i,q,k) : πq= πg ∙ πh} (i,g,j)•(j,h,k) = (i, gh, k) multiplication in G multiplication in Gijk Gk G

19. LOOP POLYGROUPOIDS REPRESENTABLE EXAMPLES

20. Z1 Z3 Z2 Z1 Z6 Z12 Multicategory (on blackboard) Z4

22. Loop polygroupoid with I={p,q,r} and Gx=Z4 and Gxy=Z1 Z1 Z1 Z1 Z1 Z4 Z4 Z4

23. Loop polygroupoid with I={p,q,r} and Gx=Z4 and Gxy=Z2 Z2 Z2 Z2 Z2 Z4 Z4 Z4

24. E E E E Point dense RA: (A, •, I, -1) E E all the groups are one-element Loop polygroupoid with I={p,q,r} and Gx=Z1 E

25. E,T E,T E,T E,T Pair dense RA: (A, •, I, -1) E,T E,T all the groups are one- or two-element E,T Loop polygroupoid with I={p,q,r,s,t} and Gx=Z2 or Gx=Z1

26. LPG structure: Group system Gxy is a common factor group of Gx and Gy Gijk Gij Hxy Gx/Hxy≡ Gy/Hyx Gx Gy Hxz Gik Gjk Group system Gxyz≡ Gx/(Hxy◦Hxz) Gi Gj A = { (i,g,j) : i,j in I and g in Gi/Hij } (i,g,j)•(j,h,k) = { (i,q,k) : q in Gi/Hik and …} Gk structure belonging to group system Gz

27. Loop polygroupoid structure Gxy is a common factor group of Gx and Gy Gijk Gij Gik Gjk Gi Gj Group system: the isomorphisms commute Theorem (G): Representable LPGs are exactly the structures belonging to group systems. Gk

28. Problem: are all LPGs representable? Partial results (AG): LPGs with I having less than 5 elements are representable. LPGs with all groups direct products of at most two finite cyclic groups are representable. LPGs with less than “three levels” are representable. Surprise (AG): There is a nonrepresentable LPG with I having 5 elements, the groups on the vertices Z2xZ2xZ2, the groups on the edges Z2xZ2, and the group in the middle Z2.

29. NONREPRESENTABLE LPG On blackboard

31. Loop polygroupoid structure in general Structure: Groups on the vertices, factor groups on the edges of the full graph on I, a group with a shift in the middle of each triangle Gxy Gxyz Cxyz Representation Theorem for LPG (AG): LPGs are exactly the structures belonging to shifted group systems. Gxz Gyz Loop polygroupoid: (A, •, I, -1) Gx Gy • partial binary operation on loops A = { (x,g,y) : x,y in I and g in Gxy } (x,g,y)•(y,h,z) = { (x, ghCxyz ,z) } element of factor group Gxyz of Gx Cxyzis called the shift in the triangle xyz Gz Conditions on next slide

33. Open Problems OProblem1. Are these all the nonrepresentable LPGs? OProblem2. Can each measurable RA be embedded into an atomic measurable RA? OProblem3. Are all representable measurable RAs completelyrepresentable?

34. The same for other structures,general systems theory