A. Romanowska and J.D.H. Smith, "Modal Theory: An Algebraic Approach to Order, Geometry and Convexity", Heldermann Verlag, Berlin, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in a certain way with the structure of approximations, we shall seek the result like the one in the second diagram in figure 3. In this case we say that P i (A) or PP i (A) is freely generated by A A with respect to the class C of monotone maps. We need two kinds of algebras defined in [29]. A bisemilattice hB; Deltai is an algebra with two semilattice operations (i.e. idempotent, commutative and associative. It is called distributive if both distributive laws hold. A left normal band hB; i is an algebra with an idempotent associative operation such that x y z = x z y. ....

....the Delta meet semilattice operation. Interpretation. U; L) V; M) min(U [ V ) max ] L [ M) and (U; L) Delta (V; M) min(U [ V ) max ] fmin(L [ M) j L 2 L; M 2 Mg) where max ] means family of maximal elements w.r.t. v ] e is interpreted as ( f;g) Theorem 6 (see also [29, 28]) PP 8 (A) is the free snack algebra generated by A. 2 Universality of PP 8 (A) Theorem 7 Let Omega be a set of operations on elements of PP 8 (A) such that is a derived operation. Then PP 8 (A) is not the free ordered Omega algebra generated by A. 2 Universality of PP 9 (A) ....

A. Romanowska and J.D.H. Smith, "Modal Theory: An Algebraic Approach to Order, Geometry and Convexity", Heldermann Verlag, Berlin, 1985.

....Another sequence of dimension discriminating equations for projective geometries was given in [9] In is not clear, however, to what extent the results of this paper can be generalized if equations of [9] are used. Finally, several algebraic models of convexity have been proposed recently, e.g. [10, 28, 30, 32]. We believe that investigation of the relationship between Carath eodory s and Helly s theorems and (dual) n distributivity in those models may lead to new intersting results. Acknowledgements: I would like to thank E.T. Schmidt and Ilya Muchnik for many fruitful discussions of the first version ....

A. Romanowska, J.D.H. Smith, "Modal Theory: An Algebraic Approach to Order, Geometry and Convexity", Heldermann Verlag, Berlin, 1985.

....one set; A A dna mix (4.3) dna ni dna family of sets; A snack (4.6) ne (4.7) ne (4.8) ne (4.9) salad (4.10) family of sets; A A dna ni dna scone (4. 9) dna dna = does not apply; ne = non existent; ni = no information (unknown) For our characterizations, we need two kinds of algebras defined in [32]. A bisemilattice hB; Deltai is an algebra with two semilattice operations, i.e. idempotent, commutative and associative. It is called distributive if both distributive laws hold. A left normal band hB; i is an algebra with an idempotent associative operation such that x y z = x z y. We ....

A. Romanowska and J.D.H. Smith, "Modal Theory: An Algebraic Approach to Order, Geometry and Convexity", Heldermann Verlag, Berlin, 1985.

....proof is not very complicated and since it exploits an unusual presentation of the equational theory, for the sake of completeness I prove the characterization theorem here. At the end of this subsection, I shall demonstrate the connection between snacks and theory of P lonka s sums of algebras [30, 25]. Definition. A snack over a poset A is a pair (U; L) where U is a finite antichain, and L = fL 1 ; L k g is a family of finite nonempty antichains satisfying the consistency condition: L i U for all i. Moreover, L itself is required to be an antichain with respect to v ] The ....

....many topics to be investigated. First, the algebraic characterization given in this paper points out to an intimate connection between these constructions and various algebras with idempotent binary operations that have been extensively studied, most notably by A. Romanowska and J.D.H. Smith, see [8, 30, 29, 28]. In [29] they characterized freely generated meet distributive bisemilattices, that is, bisemilattices satisfying only one distributive law. In [28] idempotent semirings with semilattice reducts are characterized. These algebras are closely related to scone algebras. We have seen four types of ....

A. Romanowska and J.D.H. Smith, "Modal Theory: An Algebraic Approach to Order, Geometry and Convexity", Heldermann Verlag, Berlin, 1985.

....in a certain way with the structure of approximations, we shall seek the result like the one in the second diagram in figure 3. In this case we say that P i (A) or PP i (A) is freely generated by A A with respect to the class C of monotone maps. We need two kinds of algebras defined in [22]. A bisemilattice hB; Deltai is an algebra with two semilattice (idempotent, commutative, associative) operations. It is called distributive if both distributive laws hold. A left normal band hB; i is an algebra with an idempotent associative operation such that x y z = x z y. In what ....

....e = e x = x. Order: x y iff x Delta y = x. Interpretation. U; L) V; M) min(U[V ) max ] L[M) and (U; L) Delta (V; M) min(U [V ) max ] fmin(L [M ) j L 2 L; M 2 Mg) where max ] means family of maximal elements w.r.t. v ] e is interpreted as ( f;g) Theorem 6. see also [22, 21]) PP 8 (A) is the free snack algebra generated by A. Universality of PP 8 (A) Theorem 7. Let Omega be a set of operations on PP 8 (A) such that is a derived operation. Then PP 8 (A) is not the free ordered Omega algebra generated by A. 2 Universality of PP 9 (A) Theorem ....

A. Romanowska and J.D.H. Smith. "Modal Theory: An Algebraic Approach to Order, Geometry and Convexity". Heldermann Verlag, Berlin, 1985.

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