×

One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups. (English) Zbl 1416.51004

We know from the second author [Forum Math. 30, No. 4, 973–995 (2018; Zbl 1443.06005)] that every orthomodular lattice \(X\) admits a group \(G(X)\) with a right invariant lattice order as a complete invariant. As a special case the projection lattice \(X\) of a von Neumann algebra characterizes the algebra up to \(*\)-symmetry and trivial summands of type I2. In [Stud. Log. 106, No. 1, 85–100 (2018; Zbl 1412.06011)], the second author extended the connection to \(G(X)\) to unbounded lattices. In this article, the authors extend the connection, firstly by dropping the symmetry of the orthogonality relation, and secondly they do not assume that \(X\) is a lattice. They start with a \(V\)-semilattice called a \(\perp\)-lattice. They show that the structure of a \(\perp\)-lattice is expressed by a single binary operation \(\rightarrow\) which can be conceived as quantum-logical implication. This implies that every \(\perp\)-lattice can be understood as a special type of \(L\)-algebra. The \(L\)-algebras occur in connection with lattice-ordered groups [the second author, J. Algebra 320, No. 6, 2328–2348 (2008; Zbl 1158.06009)], Garside groups [the second author, J. Algebra 439, 470–510 (2015; Zbl 1360.20030)], non-commutative prime factorization [the second author, J. Number Theory 190, 394–413 (2018; Zbl 1420.11140)] and von Neumann algebras [the second author, J. Algebra 439, 470–510 (2015; Zbl 1360.20030)]. A special class of \(L\)-algebras arises as intervals \([u-1,1]\) in right \(l\)-groups, that is, groups with a right invariant lattice order, where \(u\) is a strong order unit. These \(L\)-algebras have been called by the second author, right bricks. Therefore, the authors prove that every bounded \(\perp\)-lattice and its corresponding \(L\)-algebra is a right brick. This enables them to embed every (not necessarily bounded) \(\perp\)-lattice \(X\) into its structure group \(G(X)\). They also show that every \(\perp\)-lattice \(X\) coincides with the set of right singular elements in the positive cone of its structure group \(G(X)\) and they characterize the class of right \(*l\)-groups obtained in this way. It turns out that the bounded \(\perp\)-lattices correspond to the right \(l\)-groups with a very strong order unit. By interpreting the embedding \(1X\rightarrow G(X)\) as a universal group valued measure they prove that for a \(\perp\)-lattice \(X\), the universal property identifies group valued measures on \(X\) with group homomorphisms on the structure group \(G(X)\). Finally, using the most natural morphisms they prove that the category of \(\perp\)-lattices is equivalent to the category of right \(l\)-groups of singular type, and that both the categories are full subcategories of the category of \(L\)-algebras. In the last section of this article, spaces with a non-Hermitian sesquilinear form, quantum sets, and a new class of Garside groups are given as examples.

MSC:

51F20 Congruence and orthogonality in metric geometry
51D25 Lattices of subspaces and geometric closure systems
05B35 Combinatorial aspects of matroids and geometric lattices
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
46L51 Noncommutative measure and integration
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bachiller, D.; Cedó, F.; Vendramin, L., A characterization of finite multipermutation solutions of the Yang-Baxter equation, Publ. Mat., 62, 2, 641-649 (2018) · Zbl 1432.16030
[2] Bigard, A.; Keimel, K.; Wolfenstein, S., Groupes et anneaux réticulés, Lecture Notes in Mathematics, vol. 608 (1977), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0384.06022
[3] Boyer, S.; Gordon, C. M.; Watson, L., On L-spaces and left-orderable fundamental groups, Math. Ann., 356, 4, 1213-1245 (2013) · Zbl 1279.57008
[4] Boyer, S.; Rolfsen, D.; Wiest, B., Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble), 55, 1, 243-288 (2005) · Zbl 1068.57001
[5] Brieskorn, E.; Saito, K., Artin-Gruppen und Coxeter-Gruppen, Invent. Math., 17, 245-271 (1972) · Zbl 0243.20037
[6] Chang, C. C., Algebraic analysis of many valued logics, Trans. Amer. Math. Soc., 88, 467-490 (1958) · Zbl 0084.00704
[7] Chouraqui, F., Garside groups and Yang-Baxter equation, Comm. Algebra, 38, 12, 4441-4460 (2010) · Zbl 1216.16023
[8] Chouraqui, F., Left orders in Garside groups, Internat. J. Algebra Comput., 26, 7, 1349-1359 (2016) · Zbl 1357.20015
[9] Chouraqui, F.; Godelle, E., Finite quotients of groups of I-type, Adv. Math., 258, 46-68 (2014) · Zbl 1344.20051
[10] Clay, A.; Rolfsen, D., Ordered Groups and Topology, Graduate Studies in Mathematics, vol. 176 (2016), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1362.20001
[11] Conrad, P., Right-ordered groups, Michigan Math. J., 6, 267-275 (1959) · Zbl 0099.01703
[12] Darnel, M. R., Theory of Lattice-Ordered Groups, Monographs and Textbooks in Pure and Applied Mathematics, vol. 187 (1995), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0810.06016
[13] Dehornoy, P., Groupes de Garside, Ann. Sci. Éc. Norm. Supér. (4), 35, 2, 267-306 (2002) · Zbl 1017.20031
[14] Dehornoy, P.; Paris, L., Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. Lond. Math. Soc. (3), 79, 3, 569-604 (1999) · Zbl 1030.20021
[15] Dehornoy, P.; Digne, F.; Godelle, E.; Krammer, D.; Michel, J., (Foundations of Garside Theory. Foundations of Garside Theory, EMS Tracts in Mathematics, vol. 22 (2015), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich) · Zbl 1370.20001
[16] Deligne, P., Les immeubles des groupes de tresses généralisés, Invent. Math., 17, 273-302 (1972) · Zbl 0238.20034
[17] Dietzel, C., A right-invariant lattice-order on groups of paraunitary matrices, J. Algebra, 524, 226-249 (2019) · Zbl 1428.06002
[18] Dvurečenskij, A., Pseudo-MV algebras are intervals in \(l\)-groups, J. Aust. Math. Soc., 72, 3, 427-445 (2002) · Zbl 1027.06014
[19] Dye, H. A., On the geometry of projections in certain operator algebras, Ann. of Math., 61, 73-89 (1955) · Zbl 0064.11002
[20] Etingof, P.; Schedler, T.; Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 169-209 (1999) · Zbl 0969.81030
[21] Glass, A. M.W., Partially Ordered Groups, Series in Algebra, vol. 7 (1999), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ · Zbl 0933.06010
[22] Godelle, E., Parabolic subgroups of Garside groups, J. Algebra, 317, 1, 1-16 (2007) · Zbl 1173.20027
[23] Greechie, R. J., Orthomodular lattices admitting no states, J. Combin. Theory Ser. A, 10, 119-132 (1971) · Zbl 0219.06007
[24] Holland, S. S., Orthomodularity in infinite dimensions; a theorem of M. Solèr, Bull. Amer. Math. Soc., 32, 2, 205-234 (1995) · Zbl 0856.11021
[25] Janowitz, M. F., A note on generalized orthomodular lattices, J. Natur. Sci. Math., 8, 89-94 (1968) · Zbl 0169.02104
[26] Jans, J. P., On Frobenius algebras, Ann. of Math., 69, 392-407 (1959) · Zbl 0092.27102
[27] Kalmbach, G., Orthomodular Lattices, London Mathematical Society Monographs, vol. 18 (1983), Academic Press, Inc.: Academic Press, Inc. London · Zbl 0512.06011
[28] Lu, J.-H.; Yan, M.; Zhu, Y.-C., On the set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1, 1-18 (2000) · Zbl 0960.16043
[29] Maclagan, D.; Sturmfels, B., Introduction to Tropical Geometry, GTM, vol. 161 (2015), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1321.14048
[30] Mac Lane, S.; Moerdijk, I., Sheaves in geometry and logic, (A First Introduction to Topos Theory. A First Introduction to Topos Theory, Universitext (1994), Springer-Verlag: Springer-Verlag New York), corrected reprint of the 1992 edition · Zbl 0822.18001
[31] Mundici, D., Interpretation of AF \(C^\ast \)-algebras in Łukasiewicz sentential calculus, J. Funct. Anal., 65, 15-63 (1986) · Zbl 0597.46059
[32] Nakayama, T., On Frobeniusean algebras, I, Ann. of Math., 40, 611-633 (1939) · JFM 65.0097.04
[33] Nakayama, T., On Frobeniusean algebras, II, Ann. of Math., 42, 1-21 (1941) · JFM 67.0092.04
[34] Navas, A., On the dynamics of (left) orderable groups, Ann. Inst. Fourier (Grenoble), 60, 5, 1685-1740 (2010) · Zbl 1316.06018
[35] Navas, A.; Wiest, B., Nielsen-Thurston orders and the space of braid orderings, Bull. Lond. Math. Soc., 43, 5, 901-911 (2011) · Zbl 1235.06017
[36] Rhodes, J. B., Modular and distributive semilattices, Trans. Amer. Math. Soc., 201, 31-41 (1975) · Zbl 0326.06006
[37] Rump, W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra, 307, 1, 153-170 (2007) · Zbl 1115.16022
[38] Rump, W., \(L\)-algebras, self-similarity, and \(l\)-groups, J. Algebra, 320, 6, 2328-2348 (2008) · Zbl 1158.06009
[39] Rump, W., Semidirect products in algebraic logic and solutions of the quantum Yang-Baxter equation, J. Algebra Appl., 7, 4, 471-490 (2008) · Zbl 1153.81505
[40] Rump, W., Right \(l\)-groups, geometric Garside groups, and solutions of the quantum Yang-Baxter equation, J. Algebra, 439, 470-510 (2015) · Zbl 1360.20030
[41] Rump, W., Decomposition of Garside groups and self-similar \(L\)-algebras, J. Algebra, 485, 118-141 (2017) · Zbl 1434.20025
[42] Rump, W., The structure group of a generalized orthomodular lattice, Studia Logica, 106, 1, 85-100 (2018) · Zbl 1412.06011
[43] Rump, W., Von Neumann algebras, \(L\)-algebras, Baer \(^⁎\)-monoids, and Garside groups, Forum Math., 30, 4, 973-995 (2018) · Zbl 1443.06005
[44] Rump, W., The L-algebra of Hurwitz primes, J. Number Theory, 190, 394-413 (2018) · Zbl 1420.11140
[45] Sasaki, U., Orthocomplemented lattices satisfying the exchange axiom, J. Sci. Hiroshima Univ. (Ser. A), 17, 293-302 (1954) · Zbl 0055.25902
[46] Serre, J.-P., A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7 (1973), Springer-Verlag: Springer-Verlag New York-Heidelberg · Zbl 0256.12001
[47] Short, H.; Wiest, B., Orderings of mapping class groups after Thurston, Enseign. Math., 46, 3-4, 279-312 (2000) · Zbl 1023.57013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.