Parametric Constructive Kripke-Semantics for Standard Multi-Agent Belief and Knowledge (Knowledge As Unbiased Belief)

We propose parametric constructive Kripke-semantics for multi-agent KD45-belief and S5-knowledge in terms of elementary set-theoretic constructions of two basic functional building blocks, namely bias (or viewpoint) and visibility, functioning also as the parameters of the doxastic and epistemic accessibility relation. The doxastic accessibility relates two possible worlds whenever the application of the composition of bias with visibility to the first world is equal to the application of visibility to the second world. The epistemic accessibility is the transitive closure of the union of our doxastic accessibility and its converse. Therefrom, accessibility relations for common and distributed belief and knowledge can be constructed in a standard way. As a result, we obtain a general definition of knowledge in terms of belief that enables us to view S5-knowledge as accurate (unbiased and thus true) KD45-belief, negation-complete belief and knowledge as exact KD45-belief and S5-knowledge, respectively, and perfect S5-knowledge as precise (exact and accurate) KD45-belief, and all this generically for arbitrary functions of bias and visibility. Our results can be seen as a semantic complement to previous foundational results by Halpern et al. about the (un)definability and (non-)reducibility of knowledge in terms of and to belief, respectively

Logics of Temporal-Epistemic Actions

We present Dynamic Epistemic Temporal Logic, a framework for reasoning about operations on multi-agent Kripke models that contain a designated temporal relation. These operations are natural extensions of the well-known "action models" from Dynamic Epistemic Logic. Our "temporal action models" may be used to define a number of informational actions that can modify the "objective" temporal structure of a model along with the agents' basic and higher-order knowledge and beliefs about this structure, including their beliefs about the time. In essence, this approach provides one way to extend the domain of action model-style operations from atemporal Kripke models to temporal Kripke models in a manner that allows actions to control the flow of time. We present a number of examples to illustrate the subtleties involved in interpreting the effects of our extended action models on temporal Kripke models. We also study preservation of important epistemic-temporal properties of temporal Kripke models under temporal action model-induced operations, provide complete axiomatizations for two theories of temporal action models, and connect our approach with previous work on time in Dynamic Epistemic Logic

Relationships between Topological Properties of X and Algebraic Properties of Intermediate Rings A(X)

A topological property is a property invariant under homeomorphism, and an algebraic property of a ring is a property invariant under ring isomorphism. Let C(X) be the ring of real-valued continuous functions on a Tychonoff space X, let C*(X) ⊆ C(X) be the subring of those functions that are bounded, and call a ring A(X) an intermediate ring if C*(X) ⊆ A(X) ⊆ C(X). For a class Q of intermediate rings, an algebraic property P describes a topological property T among Q if for all A(X), B(Y) ∈ Q if A(X) and B(Y) both satisfy P, then X satisfies T if and only if Y satisfies T. An example of a topological property being described by an algebraic property among a class of intermediate rings is that of a P-space, a Tychonoff space in which every zero-set is open. We see that the property that every prime ideal of the ring is maximal describes P-spaces among rings C(X), however for the same algebraic property does not describe P-spaces among all intermediate rings. Another example of a topological property is that of an F-space, a Tychonoff space in which disjoint co-zero sets are completely separated. We see that the property that the set of prime ideals contained in a maximal ideal form a chain describes F-spaces among all intermediate rings. We investigate what other algebraic properties describe topological properties as well as other types of relationships between algebraic properties and topological properties, and we prove some theorems about how certain topological properties relate to algebraic properties of intermediate rings

Intermediate rings of complex-valued continuous functions

[EN] For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on X, let C ∗ (X, C) be its subring of bounded functions, and let Σ(X, C) denote the collection of all the rings that lie between C ∗ (X, C) and C(X, C). We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z ◦ -ideals in the rings P(X, C) in Σ(X, C) and in their real-valued counterparts P(X, C) ∩ C(X). These correlations culminate to the fact that the structure space of any such P(X, C) is βX. For any ideal I in C(X, C), we observe that C ∗ (X, C)+I is a member of Σ(X, C), which is further isomorphic to a ring of the type C(Y, C). Incidentally these are the only C-type intermediate rings in Σ(X, C) if and only if X is pseudocompact. We show that for any maximal ideal M in C(X, C), C(X, C)/M is an algebraically closed field, which is furthermore the algebraic closure of C(X)/M ∩C(X). We give a necessary and sufficient condition for the ideal CP (X, C) of C(X, C), which consists of all those functions whose support lie on an ideal P of closed sets in X, to be a prime ideal, and we examine a few special cases thereafter. At the end of the article, we find estimates for a few standard parameters concerning the zero-divisor graphs of a P(X, C) in Σ(X, C).The authors wish to thank the referee for his/her remarks which improved the paper.Acharyya, A.; Acharyya, SK.; Bag, S.; Sack, J. (2021). Intermediate rings of complex-valued continuous functions. Applied General Topology. 22(1):47-65. https://doi.org/10.4995/agt.2021.13165OJS4765221S. K. Acharyya, S. Bag, G. Bhunia and P. Rooj, Some new results on functions in C(X) having their support on ideals of closed sets, Quest. Math. 42 (2019), 1017-1090. https://doi.org/10.2989/16073606.2018.1504830S. K. Acharyya and S. K. Ghosh, On spaces X determined by the rings Ck(X) and C∞(X), J. Pure Math. 20 (2003), 9-16.S. K. Acharyya and B. Bose, A correspondence between ideals and z-filters for certain rings of continuous functions-some remarks, Topology Appl. 160 (2013), 1603-1605. https://doi.org/10.1016/j.topol.2013.06.011S. K. Acharyya and S. K. Ghosh, Functions in C(X) with support lying on a class of subsets of X, Topology Proc. 35 (2010), 127-148.S. K. Acharyya and S. K. Ghosh, A note on functions in C(X) with support lying on an ideal of closed subsets of X, Topology Proc. 40 (2012), 297-301.S. K. Acharyya, K. C. Chattopadhyay and P. Rooj, A generalized version of the rings CK(X) and C∞(X)-an enquery about when they become Noetheri, Appl. Gen. Topol. 16, no. 1 (2015), 81-87. https://doi.org/10.4995/agt.2015.3247N. L. Alling, An application of valuation theory to rings of continuous real and complexvalued functions, Trans. Amer. Math. Soc. 109 (1963), 492-508. https://doi.org/10.1090/S0002-9947-1963-0154886-0F. Azarpanah, O. A. S. Karamzadeh and A. R. Aliabad, On Z◦-ideal in C(X), Fundamenta Mathematicae 160 (1999), 15-25. https://doi.org/10.4064/fm_1999_160_1_1_15_25F. Azarpanah and M. Motamedi, Zero-divisor graph of C(X), Acta Math. Hungar. 108, no. 1-2 (2005), 25-36. https://doi.org/10.1007/s10474-005-0205-zF. Azarpanah, Algebraic properties of some compact spaces. Real Anal. Exchange 25, no. 1 (1999/00), 317-327. https://doi.org/10.2307/44153077F. Azarpanah and T. Soundararajan, When the family of functions vanishing at infinity is an ideal of C(X), Rocky Mountain J. Math. 31, no. 4 (2001), 1133-1140. https://doi.org/10.1216/rmjm/1021249434S. Bag, S. Acharyya and D. Mandal, A class of ideals in intermediate rings of continuous functions, Appl. Gen. Topol. 20, no. 1 (2019), 109-117. https://doi.org/10.4995/agt.2019.10171L. H. Byum and S. Watson, Prime and maximal ideals in subrings of C(X), Topology Appl. 40 (1991), 45-62. https://doi.org/10.1016/0166-8641(91)90057-SR. E. Chandler, Hausdorff Compactifications, New York: M. 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Characterizations of Ideals in Intermediate C

Let X be a completely regular topological space. An intermediate ring is a ring A(X) of continuous functions satisfying C*(X)⊆A(X)⊆C(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences A and ℨA are defined between ideals in A(X) and z-filters on X, and it is shown that these extend the well-known correspondences studied separately for C∗(X) and C(X), respectively, to any intermediate ring. Moreover, the inverse map A← sets up a one-one correspondence between the maximal ideals of A(X) and the z-ultrafilters on X. In this paper, we define a function A that, in the case that A(X) is a C-ring, describes ℨA in terms of extensions of functions to realcompactifications of X. For such rings, we show that ℨA← maps z-filters to ideals. We also give a characterization of the maximal ideals in A(X) that generalize the Gelfand-Kolmogorov theorem from C(X) to A(X)