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A arithmetical function f is said to be a totient if there exist completely multiplicative functions f t and f v such that f = f t ∗ f v - 1 , where ∗ is the Dirichlet convolution. Euler's ϕ -function is an important example of a totient. In this paper we find the structure of the usual product of two totients, the usual integer power of totients, the usual product of a totient and a specially multiplicative function and the usual product of a totient and a completely multiplicative function. These results are derived with the aid of generating series. We also provide some distributive-like characterizations of totients involving the usual product and the Dirichlet convolution of arithmetical functions. They give as corollaries characterizations of completely multiplicative functions. [ABSTRACT FROM AUTHOR]

Summary: In the process of evolution, the multi‐target Firefly algorithm has a single learning strategy and is easy to fall into premature convergence, resulting in poor comprehensive performance of the algorithm. To solve this problem, a multi‐objective firefly algorithm (MOFA‐LM) combining logistic mapping and Cauchy mutation is proposed in this article. To improve the uneven distribution of the initial population, the initial population with good ergodicity is generated by logistic mapping. Aiming at the problem that the population is easy to fall into local optimal, Cauchy mutation can generate relatively large disturbance when fireflies are updating their positions, which makes it easier to jump out of the local optimal value with a wide range of individual optimization, effectively improve the optimization accuracy of the algorithm, overcome the premature convergence of the algorithm and maintain the convergence of the population. In the experimental part, the proposed algorithm is compared with some multi‐objective optimization algorithms in 19 benchmark test functions, and the effectiveness of the strategy added by MOFA‐LM is verified. The results show that MOFA‐LM has obvious advantages over other algorithms in improving population convergence and distribution. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let Γ be a directed graph and Inv (Γ) be the graph inverse semigroup of Γ. Luo and Wang (Semimodularity in congruence lattice of graph inverse semigroups, 2021) showed that the congruence lattice C (Inv (Γ)) of any graph inverse semigroup Inv (Γ) is upper semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri, Mesyan and Mitchell (Properties of congruence lattices of graph inverse semigroups, 2022) characterized the directed graph Γ for which is lower semimodular. In the present paper, we show that lower semimodularity, modularity and distributivity in the congruence lattice C (Inv (Γ)) of any graph inverse semigroup Inv (Γ) are equivalent. [ABSTRACT FROM AUTHOR]

We investigate a variety of cut and choose games, their relationship with (generic) large cardinals, and show that they can be used to characterize a number of properties of ideals and of partial orders: certain notions of distributivity, strategic closure, and precipitousness. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper characterizes a uninorm distributive over a noncontinuous t-(co)norm, where the uninorm is locally internal on the boundary. Firstly, when the uninorm is disjunctive, the necessary conditions (resp. the sufficient conditions) for a uninorm distributive over a noncontinuous t-norm are analyzed under the certain condition. Secondly, the distributivity of a conjunctive uninorm over a noncontinuous t-conorm is characterized by duality. In particular, this paper is related to the open question recalled by Klement in the Linz2000 closing session, which provides the noncontinuous solutions of that question. [ABSTRACT FROM AUTHOR]

We prove “large” associativity of the partial plus operation in generalized effect algebras and present an overview of distributivity-like properties of partial operations plus and minus in generalized effect algebras with respect to (possibly infinite) suprema and infima and vice versa. These results generalize previous results in various subclasses of generalized effect algebras. [ABSTRACT FROM AUTHOR]

In Mexican Spanish, the indefinite quantifier “uno que otro” is used to refer to low cardinalities of entities separated in space or time. Even though they have a strictly singular form, noun phrases with “uno que otro” always refer to more than one entity. And, despite their notional content of plurality, they reject collective predicates. In this paper we propose a semantic analysis of this expression as a distributive indefinite, that is, an indefinite whose reference covaries as a function of a distributive operator, in the way that adnominal distributive numerals do in other languages ​​(Gil, 1983; Cable, 2014). Distributive numerals are supposed not to exist in Spanish; however, we claim that “uno que otro” is a distributed share marker (Choe, 1987; Bosnić et al 2020) related to a set of events (Balusu, 2006; Cable, 2014; Romero, 2006). This set is necessarily plural, although of low cardinality, and involves the precondition that its elements (atomic events) occur in non-contiguous time or space intervals. The semantic characterization of “uno que otro” as a distributive indefinite captures its referential dependence, its plural sense, its rejection of collective predicates and its content of separation as a secondary effect of a precondition on the events on which it is distributed. We thus show that Spanish has an adnominal expression of the same semantic nature as distributive numerals, with the precision that it requires distribution over events and not over entities.

ABSTRACT. In Mexican Spanish, the indefinite quantifier "uno que otro" is used to refer to low cardinalities of entities separated over space or time. Even though they have a strictly singular form, noun phrases with "uno que otro" always refer to more than one entity. And, despite their notional content of plurality, they reject collective predicates. In this paper we propose a semantic analysis of this expression as a distributive indefinite, that is, an indefinite whose reference covaries as a function of a distributive operator, in the way that adnominal distributive numerals do in other languages (Gil, 1983; Cable, 2014). Distributive numerals are supposed not to exist in Spanish; however, we claim that "uno que otro" is a distributed share marker (Choe, 1987; Bosnić et al 2020) related to a set of events (Balusu, 2006; Cable, 2014; Romero, 2006). This set is necessarily plural, although of low cardinality, and involves the precondition that its elements (atomic events) occur in non-contiguous time or space intervals. The semantic characterization of "uno que otro" as a distributive indefinite captures its referential dependence, its plural sense, its rejection of collective predicates and its content of separation as a secondary effect of a precondition on the events over which it is distributed. We thus show that Spanish has an adnominal expression of the same semantic nature as distributive numerals, with the precision that it requires distribution over events and not over entities. [ABSTRACT FROM AUTHOR]

This paper mainly investigates the distributivity of an extended uninorm over an extended overlap function when the uninorm is continuous in (0 , 1) 2. It first studies the distributivity of the extended representable uninorms over the extended overlap functions. Then, it presents the distributivity of the extended uninorms over the extended overlap functions when the uninorms belong to one of the usual classes U c o s , min , U c o s , max. [ABSTRACT FROM AUTHOR]

Similar to the construction of ordinal sums of overlap functions, Baczyński et al. introduced two new kinds of ordinal sums of fuzzy implications without additional restrictions on summands in 2017. In this paper, based on the ordinal sum of fuzzy implications with the first method, we discuss its basic properties, such as iterative Boolean law, right ordering property and strong boundary condition. Meanwhile, we give characterizations of such ordinal sum of fuzzy implications that is a QL -implication constructed from tuples (O , G , N ⊤) or a D -implication derived from grouping function G , and show some conclusions about its relations with (G , N) - and R O -implications. Moreover, we study the distributivity of such ordinal sum of fuzzy implications over a class of overlap and grouping functions. More specifically, we give necessary and sufficient conditions under which this ordinal sum of fuzzy implications is distributive over overlap and grouping functions satisfying some conditions. [ABSTRACT FROM AUTHOR]

The study of the universal semantic types of repetition - multiplicative, distributive, iterative on the material of the Ossetian language reveals significant discrepancies in the designation of aspectual-actional features of the course of action in mono- and polytemporal plans, especially iteration, which is, in contrast to the Russian language, in the sphere of action of a separate grammatical categories of multiplicity, and proves that different interpretations of the same objective entities by speakers of different languages is the most important characteristic of the linguistic picture of the world, which noticeably distinguishes even related languages.

This study examines a little-known case of grammaticalized distributivity in the Romance branch, as found in tenth to fifteenth century texts produced in the Iberian Peninsula. Analysis of a newly compiled, hand-annotated corpus demonstrates that, in Old Ibero-Romance, the reduplication of an adnominal cardinal numeral on the internal argument of the verb mandates a distributive reading at the sentential level. The distributive construction is characterized as 'double-object'-like structure occurring only in expressions of dynamic and static possession. Whilst a cross-categorial operation originating in the DP edge triggers pluralization of the event itself, an applicative treatment of the clausal syntax not only captures the 'double-object'-like structure of the construction, but the nature of the distributive relation itself. [ABSTRACT FROM AUTHOR]

This article describes a novel pattern of interpretations associated with universal determiners like 'each' and 'every'. It is demonstrated that these canonically distributive quantifiers can give rise to surprising collective readings when they quantify into sub-clausal constituents, especially other Determiner Phrases. For instance, 'two cards from each player' can be understood to pick out a single assorted deck of cards, one whose contents co-vary with the players. Yet this deck as a whole may be said to participate in a range of collective activities (being shuffled together, being traded en masse, not fitting into a standard pack, etc.). Such examples are shown to differ from more familiar cumulative readings of the same quantifiers. A compositional analysis is offered that generalizes Krifka's (2001) method of quantification into speech acts in order to accommodate quantification into a larger class of non-truth-denoting semantic objects, including in these cases, entities. [ABSTRACT FROM AUTHOR]

There are two cases of the (conditional) distributivity of semi-t-operators over S -uninorms that have not been discussed by Fang and Hu in 2019. This paper further characterizes the necessary condition for the left (resp. right) distributivity of semi-t-operators over S -uninorms in these two cases. The sufficient conditions for the left (resp. right) distributivity of semi-t-operators over S -uninorms are also discussed. The outcome of one case is different from the conclusion obtained by Fang and Hu, whereby it is impossible to obtain the necessary and sufficient conditions for semi-t-operators distributed over S -uninorms as usual. In particular, this paper points out that the distributivity of semi-t-operators over S -uninorms is equivalent to their conditional distributivity over S -uninorms under the two cases mentioned above. Some examples are also presented to support our conclusions. [ABSTRACT FROM AUTHOR]

English every-DPs can have cumulative readings relative to plural DPs, but only under severe syntactic constraints. This paper discusses different potential formulations of these constraints for the German correlate of every-DPs, jed-DPs. We argue that existing 'surface-oriented' hypotheses face empirical problems and propose a new hypothesis, namely that the availability of a cumulative reading relative to another plural X is sensitive to all positions in the derivational chain of X. We then spell out one possible semantic analysis of this pattern and sketch the consequences of this hypothesis for future empirical work. [ABSTRACT FROM AUTHOR]

The present article aims to develop a categorical duality for the category of finite distributive join-semilattices and ∧ -homomorphisms (maps that preserve the joins and the meets, when they exist). This dual equivalence is a generalization of the famous categorical duality given by Birkhoff for finite distributive lattices. Moreover, we show that every finite distributive semilattice is a Hilbert algebra with supremum. We obtain some applications from the dual equivalence. We provide a dual description of the 1–1 and onto ∧ -homomorphisms, and we obtain a dual characterization of some subalgebras. Finally, we present a representation for the class of finite semi-boolean algebras. [ABSTRACT FROM AUTHOR]

The goal of this article is to investigate the factors that affect the acceptability and processing of they. Previous research has sought to determine whether there are acceptability and processing differences between they/themselves with plural vs. singular antecedents, with mixed results. The studies reported here address this question using bound variable singular they (e.g., Every customer claimed that they were first in line). We asked whether bound singular they is sensitive to both the morphological number and the semantic distributivity of the binding quantifier phrase. We contrasted morphologically singular quantified antecedents (every and each) with plural quantified antecedents (all). Instead of finding an effect of number, we found an effect of semantic distributivity in acceptability, with bound singular they demonstrating a cline of preference toward more distributive antecedents. Neither number nor distributivity, however, registered as an effect on reading times. Rather, for all types of quantified antecedents, encountering a pronoun like he or she rather than they registered a processing delay, in contrast to non-quantified antecedents. Our results are most fully compatible with the view that they is underspecified for number properties. [ABSTRACT FROM AUTHOR]

The paper focuses on the semantics of distributivity, grammatical number, and cardinality predicates (numerals and modifiers like several). I argue that constructions involving so-called 'dependent plurals', i.e. plurals lacking cardinality predicates occurring in the scope of certain quantificational items such as all and most (e.g. All the girls were wearing hats), pose a challenge to familiar semantic frameworks that distinguish between two sources of multiplicity: mereological plurality and distributive quantification. I argue that dependent plural readings should be analysed as distinct both from cumulative readings and distributive readings, in the classical sense. I demonstrate how this can be accomplished in a semantic framework where expressions are evaluated relative to sets of assignments, or plural info states (van den Berg, in Stokhof and Torenvliet (eds) Proceedings of the 7th Amsterdam Colloquium, ILLC, University of Amsterdam, Amsterdam, 1990, in Dekker and Stokhof (eds) Proceedings of the 9th Amsterdam Colloquium, ILLC, University of Amsterdam, Amsterdam, 1994, Some aspects of the Internal Structure of Discourse. The Dynamics of Nominal Anaphora. PhD thesis, University of Amsterdam, 1996). The specific formal implementation is based on a modified version of Brasoveanu's (Structured nominal and modal reference. PhD thesis, Rutgers, The State University of New Jersey, 2007, Linguist Philos 31(2):129–209. https://doi.org/10.1007/s10988-008-9035-0, 2008) Plural Compositional DRT. In this framework we are able to distinguish between two types of distributivity: weak distributivity across the assignments in a single plural info state and strong distributivity across multiple info states. I argue that both of these types of distributivity play a role in the semantics of natural language, accounting for the contrasting properties of 'singular quantifiers', such as each and every, and 'plural quantifiers', such as all and most. The contrasting properties of bare plurals and plurals involving cardinality modifiers are analysed in terms of the distinction between state-level and assignment-level (mereological) plurality. [ABSTRACT FROM AUTHOR]

The (conditional) distributivity plays an important role in the construction of integrals. In this work, we aim to characterize the conditional distributivity equation only involving uninorms, where the second uninorm belongs to the well-known classes. [ABSTRACT FROM AUTHOR]

Due to the fact that Zadeh’s quantifiers constitute the usual method to modify fuzzy propositions, the so-called family of T-power based implications was proposed. In this paper, the four basic distributive laws related to T-power based fuzzy implications and fuzzy logic operations (t-norms and t-conorms) are deeply studied. This study shows that two of the four distributive laws of the T-power based implications have a unique solution, while the other two have multiple solutions. [ABSTRACT FROM AUTHOR]

The goal of this article is to investigate the factors that affect the acceptability and processing of they. Previous research has sought to determine whether there are acceptability and processing differences between they/themselves with plural vs. singular antecedents, with mixed results. The studies reported here address this question using bound variable singular they (e.g., Every customer claimed that they were first in line). We asked whether bound singular they is sensitive to both the morphological number and the semantic distributivity of the binding quantifier phrase. We contrasted morphologically singular quantified antecedents (every and each) with plural quantified antecedents (all). Instead of finding an effect of number, we found an effect of semantic distributivity in acceptability, with bound singular they demonstrating a cline of preference toward more distributive antecedents. Neither number nor distributivity, however, registered as an effect on reading times. Rather, for all types of quantified antecedents, encountering a pronoun like he or she rather than they registered a processing delay, in contrast to non-quantified antecedents. Our results are most fully compatible with the view that they is underspecified for number properties.

In this paper we provide a full characterization of functions G : [ 0 , y ¯ ] 2 → [ 0 , y ¯ ] satisfying the distributivity equation G (a ∘ b , a ∘ c) = a ∘ G (b , c) for any a , b , c with a fixed binary operation ∘ : [ 0 , y ¯ ] 2 → [ 0 , y ¯ ] generalizing minimum and maximum, where y ¯ ∈ (0 , ∞ ]. The above functional equation with an admissible function G and ∘ being minimum [resp. maximum] has appeared when studying properties of minitive [resp. maxitive] homogeneity for the recently introduced upper n -Sugeno integral. [ABSTRACT FROM AUTHOR]

In this article we will address the issue of collective nouns in European Portuguese, focusing our attention on collective nouns that denote organized groups of humans. We will begin by questioning the definition of collective nouns, introducing, for this purpose, a distinction between collective reference and a general sense of group. Then we will explore the variability that characterizes collective nouns through the application of some grammaticality tests. Further on, we will argue that a collective noun can be interpreted not only as a single collective entity, as proposed by Raposo (2013), but also as a plural entity (Ritchie 2018), which grant access to the atoms that constitute it. Finally, we will present two semantic problems involving collective nouns. First, we will discuss the relationship that can be established between collective nouns and certain proper nouns. Second, a comparison between collective nouns and some other plural expressions will be explored. In short, we will defend collective nouns as entities that, in a unique way in relation to other nouns and nominal expressions, introduce a collective reference in discourse. [ABSTRACT FROM AUTHOR]

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This paper examines Numeral-Classifier Reduplication (NCR) in Beijing Mandarin, a process that takes the numeral and classifier of the subject or object of a sentence, reduplicates it, and moves this reduplicated element between the subject and the verb. This process causes the verbal event to be distributed over (1) the associated noun as well as (2) time, such that "A person walks in" after NCR would be interpreted as "Many people walk in one after another." Previous work has not been applicable because those analyses do not give correct interpretations and do not deal with the same syntactic structure in terms of the reduplicated element. Therefore, the preliminary analysis composed of three points, based off of linguistic data elicited from a consultant of native Beijing Mandarin, is as follows. One, NCR only applies to verbal events that are bound, or have natural endpoints, which corresponds with previous work that indicate the varieties of boundedness and discreteness of verbal events. Two, NCR adjoins to the syntax similar to a manner adverbial because of their similar syntactic distribution. Three, NCR's distributive interpretation is determined by the adjunction location of the reduplicated element. When NCR attaches in the higher position, distribution can be over the subject or the object, in addition to time; in the lower position, distribution can only be over the object and time. This work constitutes one of the first in-depth look at this phenomena in Mandarin and should serve as a first step in the development of a larger research program

This article focuses on distributivity and conditional distributivity for uninorms over a given continuous t-norm, closely related to the open problem recalled by Klement in the Linz2000 closing session. This problem has been solved for the well-known classes of uninorms except the class of uninorms with the continuous underlying operators. In this article, the complete characterizations of uninorms with continuous underlying operators (conditionally) distributive over a given continuous t-norm are presented. From the obtained results, we deduce that conditional distributivity is equivalent to distributivity for a pair of such operators. [ABSTRACT FROM AUTHOR]

In this work, we analyse the internal structure and semantic properties of the Spanish non-specific complex quantifier algún que otro 'lit. some that other'. We argue that que in algún que otro and its variants (algunos que otros, uno que otro, unos que otros) is a disjunctive coordinating conjunction with an inclusive interpretation that introduces compatible alternatives, and also conveys, in combination with otro, a sense of disconnection between entities, which blocks the group interpretation and triggers a distributive reading over events. [ABSTRACT FROM AUTHOR]

In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications. And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are given. [ABSTRACT FROM AUTHOR]

In this paper, I first introduce what inalienable possession structure (IPS) is cross-linguistically as well as how to form an IPS in Mandarin Chinese, i.e., pronoun + body part or kinship term, etc. With the help of postverbal IPS, I relate the lack of plural pronominal possessor in IPS, which is never discussed in the literature, to the prohibition of distributivity over distributivity, i.e., the semantic anomaly of distributive plural possessor over the stubborn distributivity inherent to Chinese IPS nouns. I also argue that the requirement of a plural pronominal possessor seen in the IPS of public places, spatial directions, and professional titles is a result of stubborn collectivity shared by these nouns. In the end, I discuss the association between the distinction of inalienable and alienable nouns and that of active and stative verbs.

In this work, we consider a recent proposal that claims that the preferred interpretation of sentences containing definite plural expressions, such as “The boys are building a snowman,” is not determined by semantic composition but is pragmatically derived via an implicature. Plural expressions can express that each member of a group acts individually (distributive interpretation) or that the group acts together (collective interpretation). While adults prefer collective interpretations for sentences that are not explicitly marked for distributivity by the distributive marker each, children do not show this preference. One explanation is that the adult collective preference for definite plurals arises due to a conversational implicature. If implicature calculation requires memory resources, children may fail to calculate the implicature due to memory limitations. This study investigated whether loading Dutch-speaking adults' working memory, using a dual task, would elicit more child-like distributive interpretations, as would be predicted by the implicature account. We found that loading WM in adults did lead to response patterns more similar to children. We discuss whether our results offer a plausible explanation for children's development of an understanding of distributivity and how our results relate to recent debates on the role of cognitive resources in implicature calculation.