The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to "classically'' separate by hyperplanes in max-min convex geometry.
In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal $(I,f_I)$ of a partial monounary algebra $(A,f_A)$ we define the quotient partial monounary algebra $(A,f_A)/(I,f_I)$. Let $(X,f_X)$, $(Y,f_Y)$ be partial monounary algebras. We describe all partial monounary algebras $(P,f_P)$ such that $(X,f_X)$ is an ideal of $(P,f_P)$ and $(P,f_P)/(X,f_X)$ is isomorphic to $(Y,f_Y)$.
In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal lattice, we pay attention to prime, maximal, ⊙-prime ideals and to ideals that are meet-irreducible or meet-prime in the lattice of all ideals. We introduce the concept of an annihilator of a given subset of a De Morgan residuated lattice and we prove that annihilators are a particular kind of ideals. Also, regular annihilator and relative annihilator ideals are considered.
In the paper the notion of an ideal of a lattice ordered monoid A is introduced and relations between ideals of A and congruence relations on A are investigated. Further, it is shown that the set of all ideals of a soft lattice ordered monoid or a negatively ordered monoid partially ordered by inclusion is an algebraic Brouwerian lattice.
The notion of idempotent modification of an algebra was introduced by Ježek. He proved that the idempotent modification of a group is subdirectly irreducible. For an $MV$-algebra $\mathcal A$ we denote by $\mathcal A^{\prime }, A$ and $\ell (\mathcal A)$ the idempotent modification, the underlying set or the underlying lattice of $\mathcal A$, respectively. In the present paper we prove that if $\mathcal A$ is semisimple and $\ell (\mathcal A)$ is a chain, then $\mathcal A^{\prime }$ is subdirectly irreducible. We deal also with a question of Ježek concerning varieties of algebras.
We study improper interval edge colourings, defined by the requirement that the edge colours around each vertex form an integer interval. For the corresponding chromatic invariant (being the maximum number of colours in such a colouring), we present upper and lower bounds and discuss their qualities; also, we determine its values and estimates for graphs of various families, like wheels, prisms or complete graphs. The study of this parameter was inspired by the interval colouring, introduced by Asratian, Kamalian (1987). The difference is that we relax the requirement on the original colouring to be proper., Peter Hudák, František Kardoš, Tomáš Madaras, Michaela Vrbjarová., and Obsahuje seznam literatury
A sparsely encoded Willshaw-like attractor neural network based on the binary Hebbian synapses is investigated analytically and by Computer simulations. A special inhibition mechanism which supports a constant number of active neurons at each time step is used. The informationg capacity and the size of attraction basins are evaluated for the Single-Step and the Gibson-Robinson approximations, as well as for experimental results.
Non-riumerical fuzzy and possibilistic measures taking their values in
partially ordered sets, semilattices or lattices are introduced. Using the operations of supremurn and infimum in these structures, the inner and outer (lower and upper) extensions of the original measures are investigated and defined. The conditions under which the resulting functions -extend conservatively the original ones and possess the properties of fuzzy or possibilistic measures, are explicitly stated and relevant assertions are proved.
As introduced by F. Harary in 1994, a graph $ G$ is said to be an $integral$ $ sum$ $ graph$ if its vertices can be given a labeling $f$ with distinct integers so that for any two distinct vertices $u$ and $v$ of $G$, $uv$ is an edge of $G$ if and only if $ f(u)+f(v)=f(w)$ for some vertex $w$ in $G$. \endgraf We prove that every integral sum graph with a saturated vertex, except the complete graph $K_3$, has edge-chromatic number equal to its maximum degree. (A vertex of a graph $G$ is said to be {\it saturated} if it is adjacent to every other vertex of $G$.) Some direct corollaries are also presented.