Islands are combinatorial objects that can be intuitively defined on a board consisting of a finite number of cells. Based on the neighbor relation of the cells, it is a fundamental property that two islands are either containing or disjoint. Recently, numerous extremal questions have been answered using different methods. We show elementary techniques unifying these approaches. Our building parts are based on rooted binary trees and discrete geometry.Among other things, we show the maximum cardinality of islands on a toroidal board and in a hypercube. We also strengthen a previous result by rarefying the neighborhood relation.

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For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal'tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal'tsev condition. However, the Mal'tsev condition constructed in [5] is not the simplest known one in general. Now we improve this result by constructing the best Mal'tsev condition and various related conditions. As an application, we give a particularly easy new proof of the result of Freese and Jónsson [11] stating that modular congruence varieties are Arguesian, and we strengthen this result by replacing "Arguesian" by "higher Arguesian" in the sense of Haiman [18]. We show that lattice terms for congruences of an arbitrary congruence modular variety can be computed in two steps: the first step mimics the use of congruence distributivity, while the second step corresponds to congruence permutability. Particular cases of this result were known; the present approach using TIP is even simpler than the proofs of the previous partial results.

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For an arbitrary lattice identity implying modularity (or at least congruence modularity) a Mal'tsev condition is given such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal'tsev condition.It is an old problem if all congruence lattice identities are equivalent to Mal'tsev (Mal'cev) conditions. In other words, we say that a lattice identity λ can be characterized by a Mal'tsev condition if there exists a Mal'tsev condition M such that, for any variety V, λ holds in congruence lattices of all algebras in V if and only if M holds in V; and the problem is if all lattice identities can be characterized this way. This problem was raised first in Grätzer [15], where the notion of a Mal'tsev condition was defined. A strong Mal'tsev condition for varieties is a condition of the form "there exist terms h 0 , . . . , h k satisfying a set Σ of identities" where k is fixed and the form of Σ is independent of the type of algebras considered. By a Mal'tsev condition we mean a condition of the form "there exists a natural number n such that P n holds" where the P n are strong Mal'tsev conditions and P n implies P n+1 for every n. The condition "P n implies P n+1 " is usually expressed by saying that a Mal'tsev condition must be weakening in its parameter. (For a more precise definition of Mal'tsev conditions cf. Taylor [23].) The problem was repeatedly asked by several authors, including Taylor [23], Jónsson [13] and Freese and McKenzie [11].Certain lattice identities have known characterizations by Mal'tsev conditions. The first two results of this kind are Jónsson's characterization of (congruence) distributivity by the existence of Jónsson terms, cf. Jónsson [12], and Day's characterization of (congruence) modularity by the existence of Day terms, cf. Day [8]. Since Day's result will be needed in the sequel, we formulate it now. For n ≥ 2 let Presented by E. T. Schmidt.