A $(3,3)$-homogeneous quantum logic with $18$ atoms. I

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.