Sets, lattices, and Boolean algebras
About: This article is published in American Mathematical Monthly.The article was published on 1970-05-01. It has received 85 citations till now.
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TL;DR: The distinction between mass nouns and count nouns, first remarked upon by Jespersen (1909, vol. 2, eh.5) in connection with English, is found in a number of the world's languages, including Chinese, Tamil, German and French.
Abstract: The distinction between mass nouns and count nouns, first remarked upon by Jespersen (1909, vol. 2, eh. 5.2) in connection with English, is found in a number of the world's languages, including Chinese, Tamil, German and French. In English, the most common way to distinguish these two classes of words is syntactic. Cardinal numerals and quasi-cardinal numerals (e.g., \"several\") modify count nouns, never mass nouns. Moreover, \"little\" and \"much\" modify mass nouns, never count nouns; whereas '\"few\" and \"many\" modify count nouns, never mass nouns. Count nouns admit a morphological contrast between singular and plural; mass nouns do not, being almost always singular. The pronoun \"'one\" may have as its antecedent a count noun, not a mass noun (Baker 1978, ch. 10.1). Mass nouns with singular morphology do not tolerate the indefinite article, whereas singular count nouns do. Finally, mass nouns occur only with the plural form of those quantifiers whose singular and plural forms differ. It has also been thought that mass nouns and count nouns can be distinguished by what they denote. The two criteria most commonly proposed are: cumulativity and divisivity of reference. Quine (1960, p. 91) observed that if a mass term such as \"water\" is true of each of two items then it is true of the two items taken together; and he dubbed this seman-tical property of mass terms '\"cumulative reference\". This characterization , while apt, does not, however, distinguish mass nouns from count nouns; for, as Link (1991, pp. 4-5) has pointed out, cumulativity of reference also holds of plural count nouns: Just as it is the case that \"If the animals in this camp are horses and the animals in that camp are horses, then the animals in the two camps are horses\"; so it is the case thank the audiences in attendance at these presentations for their comments and criticisms. t also thank Peter Simons for written comments on an earlier draft of this paper.
208 citations
TL;DR: For a commutative ring R with set of zero-divisors Z (R), the zero-Divisor graph of R is Γ( R ) = Z ( R )−{0), with distinct vertices x and y adjacent if and only if xy = 0 as mentioned in this paper.
194 citations
TL;DR: This paper introduces design spaces that model physical connectivity, functionality, and assemblability considerations for a representative product family, a class of coffeemakers, and demonstrates how these spaces can be combined into a “common” product variety design space.
Abstract: For typical optimization problems, the design space of interest is well defined: It is a subset of Rn, where n is the number of (continuous) variables. Constraints are often introduced to eliminate infeasible regions of this space from consideration. Many engineering design problems can be formulated as search in such a design space. For configuration design problems, however, the design space is much more difficult to define precisely, particularly when constraints are present. Configuration design spaces are discrete and combinatorial in nature, but not necessarily purely combinatorial, as certain combinations represent infeasible designs. One of our primary design objectives is to drastically reduce the effort to explore large combinatorial design spaces. We believe it is imperative to develop methods for mathematically defining design spaces for configuration design. The purpose of this paper is to outline our approach to defining configuration design spaces for engineering design, with an emphasis on the mathematics of the spaces and their combinations into larger spaces that more completely capture design requirements. Specifically, we introduce design spaces that model physical connectivity, functionality, and assemblability considerations for a representative product family, a class of coffeemakers. Then, we show how these spaces can be combined into a “common” product variety design space. We demonstrate how constraints can be defined and applied to these spaces so that feasible design regions can be directly modeled. Additionally, we explore the topological and combinatorial properties of these spaces. The application of this design space modeling methodology is illustrated using the coffeemaker product family.
71 citations
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TL;DR: In this article, the concept of soft sets was applied to the theory of subtraction algebras, and the notion of soft WS-algebra, soft subalgebra and soft deductive systems were derived.
Abstract: Molodtsov [8] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to the theory of subtraction algebras. The notion of soft WS-algebras, soft subalgebras and soft deductive systems are introduced, and their basic properties are derived.
53 citations
Cites background from "Sets, lattices, and Boolean algebra..."
...The ordered set (X;≤) is a semi-Boolean algebra in the sense of [1], that is, it is a meet semilattice with zero 0 in which every interval [0, a] is a Boolean algebra with respect to the induced order....
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18 Aug 1996
TL;DR: A Product Module Reasoning System (PMRS) is developed to reason about sets of product architectures, to translate design requirements into constraints on these sets, and to directly enumerate all feasible modules without generate-and-test or heuristic search approaches.
Abstract: A product’s architecture affects the ability of a company to customize, assemble, service, and recycle the product. Much of the flexibility to address these issues is locked into the product’s design during the configuration design stage when the architecture is determined. The concepts of modules and modularity are central to the description of an architecture, where a module is a set of components that share some characteristic. Modularity is a measure of the correspondence between the modules of a product from different viewpoints, such as functionality and physical structure. The purpose of this paper is to investigate formal foundations for configuration design. Since product architectures are discrete structures, discrete mathematics, including set theory and combinatorics, is used for the investigation. A Product Module Reasoning System (PMRS) is developed to reason about sets of product architectures, to translate design requirements into constraints on these sets, to compare architecture modules from different viewpoints, and to directly enumerate all feasible modules without generate-and-test or heuristic search approaches. The PMRS is described mathematically and applied to the design of architectures for a hand-held tape recorder. Life cycle requirements are used as design criteria.
48 citations