One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups
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.
scientific article; zbMATH DE number 7040784
scientific article