Abstract. The consistency problem for a class of algebraic structures asks for an algorithm to decide for any given conjunction of equations whether it admits a non-trivial satisfying assignment within some member of the class. By Adyan (1955) and Rabin (1958) it is known unsolvable for (the class of) groups and, recently, by Bridson and Wilton (2015) for finite groups. We derive unsolvability for (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite dimensional vector spaces over a fixed or arbitrary field of characteristic 0. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann-Cayley algebra and to functional and embedded multivalued dependencies in databases.

A dyadic subbase S of a topological space X is a subbase consisting of a countable collection of pairs of open subsets that are exteriors of each other. If a dyadic subbase S is proper, then we can construct a dcpo DS in which X is embedded. We study properties of S with respect to two aspects. (i) Whether the dcpo DS is consistently complete depends on not only S itself but also the enumeration of S. We give a characterization of S that induces the consistent completeness of DS regardless of its enumeration. (ii) If the space X is regular Hausdorff, then X is embedded in the minimal limit set of DS. We construct an example of a Hausdorff but non-regular space with a dyadic subbase S such that the minimal limit set of DS is empty.

We construct oriented matroids of rank 3 on 13 points whose realization spaces are disconnected. They are defined on smaller points than the known examples with this property. Moreover, we construct the one on 13 points whose realization space is a connected and non-irreducible semialgebraic variety. Oriented Matroids and MatricesThroughout this section, we fix positive integers r and n.Let X = (x 1 , . . . , x n ) ∈ R rn be a real (r, n) matrix of rank r, and E = {1, . . . , n} be a set of labels of the columns of X. For such matrix X, a map χ X can be defined as χ X : E r → {−1, 0, +1}, χ X (i 1 , . . . , i r ) := sgn det(x i1 , . . . , x ir ).The map χ X is called the chirotope of X. The chirotope χ X encodes the information on the combinatorial type which is called the oriented matroid of X. In this case, the oriented matroid determined by χ X is of rank r on E.We note for some properties which the chirotope χ X of a matrix X satisfies.

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