Abstract. Rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. By a patch lattice we mean a rectangular lattice whose weak corners are coatoms. As a sort of gluings, we introduce the concept of a patchwork system. We prove that every glued sum indecomposable planar semimodular lattice is a patchwork of its maximal patch lattice intervals "sewn together"; see Figure 3 for a first impression. For a modular planar lattice, our patchwork system coincides with the S-glued system introduced by C. Herrmann in 1973. Among planar semimodular lattices, patch lattices are characterized as the patchwork-irreducible ones. They are also characterized as the indecomposable ones with respect to the Hall-Dilworth gluing over chains; this fact gives another structure theorem for planar semimodular lattices since patch lattices are obtained from the four-element non-chain lattice by adding forks, introduced in our preceding paper.

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Abstract. For subnormal subgroups A B and C D of a given group G, the factor B/A will be called subnormally down-and-up Clearly, B/A ∼ = D/C in this case. As G. Grätzer and J. B. Nation [6] have just pointed out, the standard proof of the classical Jordan-Hölder theorem yields somewhat more than widely known; namely, the factors of any two given composition series are the same up to subnormal down-and-up projectivity and a permutation. We prove the uniqueness of this permutation.The main result is the analogous statement for semimodular lattices. Most of the paper belongs to pure lattice theory; the group theoretical part is only a simple reference to a classical theorem of H. Wielandt [14].

Let F be a finite set of circles in the plane. We point out that the usual convex closure restricted to F yields a convex geometry, that is, a combinatorial structure introduced by P. H. Edelman in 1980 under the name "anti-exchange closure system". We prove that if the circles are collinear and they are arranged in a "concave way", then they determine a convex geometry of convex dimension at most 2, and each finite convex geometry of convex dimension at most 2 can be represented this way. The proof uses some recent results from Lattice Theory, and some of the auxiliary statements on lattices or convex geometries could be of separate interest. The paper is concluded with some open problems.

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Let L be a join-distributive lattice with length n and width(Ji L) ≤ k. There are two ways to describe L by k − 1 permutations acting on an n-element set: a combinatorial way given by P. H. Edelman and R. E. Jamison in 1985 and a recent lattice theoretical way of the second author. We prove that these two approaches are equivalent. Also, we characterize join-distributive lattices by trajectories.

For a lattice L, let Princ(L) denote the ordered set of principal congruences of L. In a pioneering paper, G. Grätzer characterized the ordered sets Princ(L) of finite lattices L; here we do the same for countable lattices. He also showed that each bounded ordered set H is isomorphic to Princ(L) of a bounded lattice L. We prove a related statement: if an ordered set H with least element is the union of a chain of principal ideals, then H is isomorphic to Princ(L) of some lattice L.

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