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Brief communications
A $(3,3)$-homogeneous quantum logic with $18$ atoms. I
F. F. Sultanbekov Chair of Mathematical Analysis, Kazan (Volga Region) Federal University, Kazan, Russia
Abstract:
A quantum logic is called
$(m,n)$-homogeneous if any its atom is contained exactly in
$m$ maximal (with respect to inclusion) orthogonal sets of atoms (we call them
blocks), and every block contains exactly
$n$ elements. We enumerate atoms by natural numbers. For each block
$\{i,j,k\}$ we use the abbreviation
$i$-
$j$-
$k$. Every such logic has the following
$7$ initial blocks
$B_1,\dots,B_7$:
$1$-
$2$-
$3$,
$1$-
$4$-
$5$,
$1$-
$6$-
$7$,
$2$-
$8$-
$9$,
$2$-
$10$-
$11$,
$3$-
$12$-
$13$, and
$3$-
$14$-
$15$. For an
$18$-atom logic the arrangements of the rest atoms
$16,17$, and
$18$ is important. We consider the case when they form a loop of order
$4$ in one of layers composed of initial blocks, for example,
$l_4$:
$3$-
$14$-
$15$,
$15$-
$16$-
$17$,
$17$-
$18$-
$13$, and
$13$-
$12$-
$3$. We prove that there exist (up to isomorphism) only
$5$ such logics, and describe pure states and automorphism groups for them.
Keywords:
quantum logic, homogeneous quantum logic, $(3,3)$-homogeneous logic, atom, block, pure state, automorphism group.
UDC:
512 Presented by the member of Editorial Board: D. Kh. MushtariReceived: 22.05.2012
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