Quantum Matroids

Abstract

We define a quantum matroid to be any finite nonempty poset $P$ satisfying the conditions R, SL, M, AU below.

R: $P$ is ranked.

SL: $P$ is a (meet) semilattice.

M: For all $x \in P$, the interval $[0, x]$ is a modular atomic lattice.

AU: For all $x, y \in P$ satisfying $\operatorname{rank}(x) < \operatorname{rank}(y)$, there exists an atom $a \in P$ such that $a \le y$, $a \not\le x$, and such that $x \lor a$ exists in $P$.

Condition AU is the augmentation axiom.

We develop a theory of quantum matroids. Although we deal at length with the general case, our emphasis is on quantum matroids $P$ with the following extra structure: We say $P$ is nontrivial if $P$ has rank $D \ge 2$, and $P$ is not a modular atomic lattice. In what follows suppose $P$ is nontrivial. We say $P$ is $q$-line regular whenever each rank 2 element in $P$ covers exactly $q + 1$ elements of $P$. We say $P$ is $\beta$-dual-line regular whenever each element in $P$ with rank $D - 1$ is covered by exactly $\beta + 1$ elements of $P$. We say $P$ is $\alpha$-zig-zag regular whenever for all pairs $x, y \in P$ such that $\operatorname{rank}(x)=D-1$, $\operatorname{rank}(y)=D$, and such that $x$ covers $x \wedge y$, there exists exactly $\alpha + 1$ pairs $x', y' \in P$ such that $y'$ covers $x$, $y'$ covers $x'$, and such that $y$ covers $x'$. We say $P$ is regular whenever $P$ is line regular, dual-line regular, and zig-zag regular. We prove the following theorem.

Theorem. Let $D$ denote an integer at least 4. Then a poset $P$ is a nontrivial regular quantum matroid of rank $D$ if and only if $P$ is isomorphic to one of the following:

(i) A truncated Boolean algebra $B(D, N)$, $(D < N)$.

(ii) A Hamming matroid $H(D, N)$, $(2 \le N)$.

(iii) A truncated projective geometry $L_q(D, N)$, $(D < N)$.

(iv) An attenuated space $A_q(D, N)$, $(D < N)$.

(v) A classical polar space of rank $D$.