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We define an analogue of Schnyder's tree decompositions for 3-connected planar graphs. Based on this structure we obtain: • Let G be a 3-connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f−1)×(f−1) grid. • Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3. The second result...
Let I, H, S, P be the usual class operators on universal algebras. For a class K of universal algebras of the same type, let R({K}) be the class of all algebras isomorphic to a retract of a member of K and let R denote the corresponding class operator. In this paper the semigroup generated by class operators I, R, H, S, P and the corresponding partially ordered set are described. Also the standard...
An n-ary operation f is totally symmetric if it obeys the identity f(x1,...,xn)=f(y1,...,yn) for all sets of variables such that {x1,...,xn}={y1,...,yn}. We characterize finite posets admitting an n-ary idempotent totally symmetric operation for all n. The characterization is expressed in terms of zigzags, special objects related to the poset. Some open problems concerning idempotent...
In this paper we introduce a new version of the concept of order varieties. Namely, in addition to closure under retracts and products we require that the class of posets should be closed under taking idempotent subalgebras. As an application we prove that the variety generated by an order-primal algebra on a finite connected poset P is congruence modular if and only if every idempotent subalgebra...
We consider minimal interval extensions of a partial order which preserve the height of each vertex. We show that minimal interval extensions having this property bijectively correspond to the maximal chains of a sublattice of the lattice of maximal antichains of the given order. We show that they also correspond to the set of minimal interval extensions of a certain extension of this order.
Given two closure spaces (E,ϕ) and (E′,ϕ′), a relation R⊒E×E′ is said biclosed if every row of its matrix representation corresponds to a closed subset of E′, and every column to a closed subset of E. An isomorphism between, on the one hand, the set of all biclosed relations and, on the other hand, the set of all Galois connections between the two lattices of closed sets is established. Several computational...
In this paper we study the role of cleavability and divisibility in the topology of generalized ordered (GO-)spaces. We characterize cleavability of a GO-space over the class of metrizable spaces, and over the spaces of irrational and rational numbers. We present a series of examples related to characterizations of cleavability over separable metric spaces and over the space of real numbers.
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