Geodesic
A $(3,3)$-homogeneous quantum logic with~$18$ atoms.~I
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2012), pp. 72-78.

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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
Mots-clés : automorphism group. 
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F. F. Sultanbekov. A $(3,3)$-homogeneous quantum logic with~$18$ atoms.~I. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2012), pp. 72-78. http://geodesic.mathdoc.fr/item/IVM_2012_11_a6/

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