complete distributivity
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2016 ◽  
Vol 09 (04) ◽  
pp. 1650088
Author(s):  
Ivan Chajda ◽  
Helmut Länger

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.



2015 ◽  
Vol 65 (1) ◽  
Author(s):  
Judita Lihová ◽  
Ján Jakubík

AbstractThe notion of torsion class of abelian cyclically ordered groups has been introduced and fundamental properties of the collection T of all such classes, ordered by the class-theoretical inclusion, have been proved by the second author in 2011. The present paper can be considered as a continuation of the above mentioned one. We describe all atoms of T , show that T does not have any dual atom and prove complete distributivity of T .



2010 ◽  
Vol 20 (08) ◽  
pp. 969-975 ◽  
Author(s):  
U. M. SWAMY ◽  
B. VENKATESWARLU

α-Irreducible and α-Strongly Irreducible Ideals of a ring have been characterized in [2] and [4]. A complete lattice which is generated by compact elements is called an algebraic lattice for the simple reason that every such lattice is isomorphic to the lattice of subalgebras of a suitable universal algebra and vice-versa. In this paper, we characterize the irreducible elements and strongly irreducible elements in an algebraic lattice, which extends the results in [4] to arbitrary algebraic lattices. Also we obtain certain necessary and sufficient conditions, in terms of irreducible elements, for an algebraic lattice to satisfy the complete distributivity.



2003 ◽  
Vol 128 (4) ◽  
pp. 401-410
Author(s):  
Jiří Rachůnek


2000 ◽  
Vol 10 (6) ◽  
pp. 719-745 ◽  
Author(s):  
MICHAEL HUTH ◽  
ACHIM JUNG ◽  
KLAUS KEIMEL

We study continuous lattices with maps that preserve all suprema rather than only directed ones. We introduce the (full) subcategory of FS-lattices, which turns out to be *-autonomous, and in fact maximal with this property. FS-lattices are studied in the presence of distributivity and algebraicity. The theory is extremely rich with numerous connections to classical Domain Theory, complete distributivity, Topology and models of Linear Logic.



1994 ◽  
Vol 2 (2) ◽  
pp. 119-144 ◽  
Author(s):  
Robert Rosebrugh ◽  
R. J. Wood


1992 ◽  
Vol 35 (4) ◽  
pp. 537-547 ◽  
Author(s):  
Robert Rosebrugh ◽  
R. J. Wood

AbstractA complete lattice L is constructively completely distributive,(CCD)(L), if the sup map defined on down closed subobjects has a left adjoint. We characterize preservation of this property by left exact functors between toposes using a "logical comparison transformation". The characterization is applied to (direct images of) geometric morphisms to show that local homeomorphisms (in particular, product functors) preserve (CCD) objects, while preserving (CCD) objects implies openness.



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