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LetG be a closure system on a setA, and let ℌ ⊂G be a closure system consisting of allg-closed elements ofG, whereg is a family of unary operations onA. In this note we discuss whether ℌ is a closed sublattice ofG-this means, whether arbitrary suprema taken in ℌ, and taken inG, coincide. An example shows that ℌ is not a closed sublattice ofG in general. We give in theorem 2 a necessary and sufficient condition ofG that all unary-defined ℌ ⊂G are closed sublattices ofG. But of course very fewG fulfill this condition. For an arbitrary closure-systemG we get with the aid of the concept of ‘relative character’ and of theorem 1: IfG is of character ≤k then (theorem 3) a unary-defined ℌ is a closed sublattice ofG iff thek-small suprema of ℌ and ofG coincide. The fact that a closure-system is of character ≤k iff it is obtainable as the subalgebra-system of an at mostk-ary universal algebra now implies: IfG is the subalgebra-system Sub (A, f) of a universal algebra (A, f) then (theorem, 4) ℌ is a closed sublattice ofG iff ℌ is definable by endomorphisms of (A, f). If one defines automorphisms of a closure system in an obvious way, theorem 5 gives another sufficient condition for ℌ to be a closed sublattice ofG.

A very simple proof of the finite embeddability property for residuated distributive-lattice-ordered groupoids and some related classes of structures is presented. In particular, this gives an answer to the question, posed by Blok and van Alten, whether the class of residuated ordered groupoids has the property. The presented construction improves the computational-complexity upper bound of the universal theory of residuated distributive-lattice-ordered groupoids given by Buszkowski and Farulewski; for chains in the class, a tight bound is obtained.

I think it is hopeless to try to give a concise account of the over 300 publications of George Grätzer and E. Tamás Schmidt. There is a danger of getting lost in too many details (for instance, mentioning Grätzer [G35], where the concept of Mal’cev condition was introduced and named). Hence, I will be guided by the following five restrictions. (1) I will discuss only lattice theoretic results. (2) I will try to concentrate on their deepest results, (3) on results with the largest impact, and (4) on series of papers. (5) I will give preference to joint or at least partially joint works. Even then, I will be able to cover only a small part of their results satisfying these five criteria.

We characterize the injective and projective regular double Stone algebras, and describe those regular double Stone algebras which are also projective in the category of double Stone algebras.

We present a new embedding of a finite join-semidistributive lattice into a finite atomistic join-semidistributive lattice. This embedding turns out to be the largest extension, when applied to a finite convex geometry.

A spectral space is localic if it corresponds to a frame under Stone Duality. This class of spaces was introduced by the author (under the name ’locales’) as the topological version of the classical frame theoretic notion of locales, see Johnstone and also Picado and Pultr). The appropriate class of subspaces of a localic space are the localic subspaces. These are, in particular, spectral subspaces. The following main questions are studied (and answered): Given a spectral subspace of a localic space, how can one recognize whether the subspace is even localic? How can one construct all localic subspaces from particularly simple ones? The set of localic subspaces and the set of spectral subspaces are both inverse frames. The set of localic subspaces is known to be the image of an inverse nucleus on the inverse frame of spectral subspaces. How can the inverse nucleus be described explicitly? Are there any special properties distinguishing this particular inverse nucleus from all others? Colimits of spectral spaces and localic spaces are needed as a tool for the comparison of spectral subspaces and localic subspaces.