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For $\\kappa $ a regular uncountable cardinal, we show that distributivity and base trees for $P(\\kappa )/{<}\\kappa $ of intermediate height in the cardinal interval $[\\omega , \\kappa )$ exist in certain models. We also show that base trees of height $\\kappa $ can exist as well as base trees of various heights $\\geq \\kappa ^+$ depending on the spectrum of cardinalities of towers in $P(\\kappa )/{<}\\kappa $ .

Inspired by the thought of distributivity between semi-t-operator and S-uninorm, this paper primarily explores the distributivity between extended semi-t-operator and extended S-uninorm on fuzzy truth value. First, Zadeh-extended semi-t-operator and S-uninorm are proposed on fuzzy truth value and some results of extended semi-t-operator are studied under special fuzzy truth values. Then, it concentrates on the sufficient condition about left and right distributivity of extended semi-t-operator over extended S-uninorm under the condition that semi-t-operator is left and right distributive over S-uninorm, respectively. Finally, when parameters satisfy different cases, sufficient conditions for the distributivity between extended semi-t-operator and extended S-uninorm are given under the condition that semi-t-operator satisfies distributivity or conditional distributivity over S-uninorm.

Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\\infty$-)categories of spans (or correspondences). In this paper, we study the more complicated setup where we have two pushforwards (an ‘additive’ and a ‘multiplicative’ one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\\infty$-categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading ‘monoid-like’ structures to ‘ring-like’ ones. For example, symmetric monoidal $\\infty$-categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$-sets, extend the norms for finite étale maps in motivic spectra to a functor from certain bispans in schemes, and make $\\mathrm {Perf}(X)$ for $X$ a spectral Deligne–Mumford stack a functor of bispans using a multiplicative pushforward for finite étale maps in addition to the usual pullback and pushforward maps. Combining this with the polynomial functoriality of $K$-theory constructed by Barwick, Glasman, Mathew, and Nikolaus, we obtain norms on algebraic $K$-theory spectra.

A classical theorem of Balcar, Pelant, and Simon says that there is a base matrix of height ${\\mathfrak h}$ , where ${\\mathfrak h}$ is the distributivity number of ${\\cal P} (\\omega ) / {\\mathrm {fin}}$ . We show that if the continuum ${\\mathfrak c}$ is regular, then there is a base matrix of height ${\\mathfrak c}$ , and that there are base matrices of any regular uncountable height $\\leq {\\mathfrak c}$ in the Cohen and random models. This answers questions of Fischer, Koelbing, and Wohofsky.

Abstract Location selection is a critical process in decision-making for projects that involve multiple criteria, such as urban planning, industrial site development, or green building projects. Multiple criteria decision making (MCDM) is a systematic approach that evaluates and ranks potential alternatives based on a set of often conflicting criteria. This study focuses on selecting the optimal urban location for a green building project by employing theL_(q)*L q ∗ q-rung orthopair multi-fuzzy soft-MCDM(L_(q)*L q ∗ q-ROMFS) techniques. TheL_(q)*L q ∗ q-ROMFS set combines elements from two distinct theories with lattice ordering parameters: q-rung orthopair fuzzy set and multi-fuzzy soft set. It provides a mathematical framework with multiple parameters that effectively represents problems involving multi-dimensional data within a dataset. We expand this concept by establishing the algebraic structures ofL_(q)*L q ∗ q-ROMFS sets, including properties like modularity and distributivity, while also analyzing their homomorphism under lattice mappings. Finally, leveraging theL_(q)*L q ∗ q-ROMFS matrix, we propose both a choice matrix and a weighted choice matrix to effectively address the selection of the optimal urban location for a green building project.

Epistemic modals have peculiar logical features that are challenging to account for in a broadly classical framework. For instance, while a sentence of the form p ∧ ◊ ¬ p (‘ p , but it might be that not  p ’) appears to be a contradiction, ◊ ¬ p does not entail ¬ p , which would follow in classical logic. Likewise, the classical laws of distributivity and disjunctive syllogism fail for epistemic modals. Existing attempts to account for these facts generally either under- or over-correct. Some theories predict that p ∧ ◊ ¬ p , a so-called epistemic contradiction , is a contradiction only in an etiolated sense, under a notion of entailment that does not always allow us to replace p ∧ ◊ ¬ p with a contradiction; these theories underpredict the infelicity of embedded epistemic contradictions. Other theories savage classical logic, eliminating not just rules that intuitively fail, like distributivity and disjunctive syllogism, but also rules like non-contradiction, excluded middle, De Morgan’s laws, and disjunction introduction, which intuitively remain valid for epistemic modals. In this paper, we aim for a middle ground, developing a semantics and logic for epistemic modals that makes epistemic contradictions genuine contradictions and that invalidates distributivity and disjunctive syllogism but that otherwise preserves classical laws that intuitively remain valid. We start with an algebraic semantics , based on ortholattices instead of Boolean algebras, and then propose a possibility semantics , based on partial possibilities related by compatibility. Both semantics yield the same consequence relation, which we axiomatize. We then show how to lift an arbitrary possible worlds model for a non-modal language to a possibility model for a language with epistemic modals. The goal throughout is to retain what is desirable about classical logic while accounting for the non-classicality of epistemic vocabulary.

In the theory of nonadditive integrals, an indispensable step is to define a pair of pseudo-addition and pseudo-multiplication that fulfill the conditional distributivity, leading to a structure of an ordered semiring in some sense. In this paper, we focus on conditionally distributive uninorms locally internal on the boundary, show that the second involved uninorm must be locally internal, and present a general framework of structures of such a pair of uninorms.

Although the scholars studied the distributivity for S-uninorms, at least, the underlying uninorm of one S-uninorm in the distributivity equations was assumed to be in Umin except the distributivity for S-uninorms over t-(co)norms. In this paper, we further characterize the distributivity for S-uninorms, where the conjunctive underlying uninorms of the S-uninorms in the distributivity equations are not fixed in Umin but arbitrary. Firstly, we discuss the distributivity between S-uninorms. Secondly, we analyze the distributivity for S-uninorms over T-uninorms. Moreover, we obtain the distributivity for T-uninorms over S-uninorms by duality. Thirdly, we investigate the distributivity for S-uninorms over disjunctive uninorms. Because S-uninorms in those distributivity equations are arbitrary, our results are extensions of the previous results on the distributivity for S-uninorms.

This paper investigates nga-marked numerals in Albanian. They qualify as distributive numerals, since the presence of nga on the numeral yields a distributive reading of the sentences they belong to. Beyond their differences, most of the previous accounts rely on the hypothesis that distributive numerals introduce some kind of semantic feature, e.g. a covariation feature; an evaluation plurality requirement, also called a post-suppositional plurality requirement; or a distributivity force. Our main claim goes against this trend of thinking. We propose that distributive numerals do not carry any semantic feature but only a formal syntactic feature that needs to enter a syntactic dependency relation with a distributivity feature. The analysis is implemented in terms of Zeijlstra’s (2004) upward agree.

Location selection is a critical process in decision-making for projects that involve multiple criteria, such as urban planning, industrial site development, or green building projects. Multiple criteria decision making (MCDM) is a systematic approach that evaluates and ranks potential alternatives based on a set of often conflicting criteria. This study focuses on selecting the optimal urban location for a green building project by employing the L q ∗ q-rung orthopair multi-fuzzy soft-MCDM( L q ∗ q-ROMFS) techniques. The L q ∗ q-ROMFS set combines elements from two distinct theories with lattice ordering parameters: q-rung orthopair fuzzy set and multi-fuzzy soft set. It provides a mathematical framework with multiple parameters that effectively represents problems involving multi-dimensional data within a dataset. We expand this concept by establishing the algebraic structures of L q ∗ q-ROMFS sets, including properties like modularity and distributivity, while also analyzing their homomorphism under lattice mappings. Finally, leveraging the L q ∗ q-ROMFS matrix, we propose both a choice matrix and a weighted choice matrix to effectively address the selection of the optimal urban location for a green building project.