We identify two key conditions that a subset A of a poset P may satisfy to guarantee the transfer of continuity properties from P to A. We then highlight practical cases where these key conditions ...are fulfilled. Along the way we are led to consider subsets of a given poset P whose way-below relation is the restriction of the way-below relation of P, which we call way-below preserving subposets. As an application, we show that every conditionally complete poset with the interpolation property contains a largest continuous way-below preserving subposet. Most of our results are expressed in the general setting of Z theory, where Z is a subset system.
We show that for every orthomodular poset P=(P,≤,′,0,1) of finite height there can be defined two operators forming an adjoint pair with respect to an order-like relation defined on the power set of ...P. This enables us to introduce the so-called operator residuated poset corresponding to P from which the original orthomodular poset P can be recovered. We show that this construction of operators can be applied also to so-called weakly orthomodular and dually weakly orthomodular posets. Examples of such posets are included.
Residuation in finite posets Chajda, Ivan; Länger, Helmut
Mathematica Slovaca,
08/2021, Letnik:
71, Številka:
4
Journal Article
Recenzirano
Odprti dostop
When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce implication to be everywhere defined and satisfying (left) adjointness ...with conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.
In this paper, we consider a common generalization of both s2-continuous posets and quasicontinuous domains, and we introduce new concepts of way below relations and s2-quasicontinuous posets. The ...main results are: (1) The way below relation on an s2-quasicontinuous poset has the interpolation property; (2) The λ2-topology on an s2-quasicontinuous poset is completely regular; (3) A poset is s2-continuous iff it is meet s2-continuous and s2-quasicontinuous.
In order to be able to use methods of universal algebra for investigating posets, we assigned to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally ...pseudocomplemented poset, a certain algebra (based on a commutative directoid or on a λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and λ-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.
Felsner, Li and Trotter showed that the dimension of the adjacency poset of an outerplanar graph is at most 5, and gave an example of an outerplanar graph whose adjacency poset has dimension 4. We ...improve their upper bound to 4, which is then best possible.
The rank-generating functions of upho posets Gao, Yibo; Guo, Joshua; Seetharaman, Karthik ...
Discrete mathematics,
January 2022, 2022-01-00, Letnik:
345, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Upper homogeneous finite type (upho) posets are a large class of partially ordered sets with the property that the principal order filter at every vertex is isomorphic to the whole poset. Well-known ...examples include k-ary trees, the grid graphs, and the Stern poset. Very little is known about upho posets in general. In this paper, we construct upho posets with Schur-positive Ehrenborg quasisymmetric functions, whose rank-generating functions have rational poles and zeros. We also categorize the rank-generating functions of all planar upho posets. Finally, we prove the existence of an upho poset with an uncomputable rank-generating function.
We show that every poset P with an antitone involution can be extended to a commutative integral residuated poset E(P). If, moreover, P is a lattice then so is E(P).
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