Information: - Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. It is the scientific and practical approach to computation and its applications and the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms) that underlie the acquisition, representation, processing, storage, communication of, and access to information. An alternate, more succinct definition of computer science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems. - In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by or , as the identity element. The free monoid on a set "A" is usually denoted "A". The free semigroup on "A" is the subsemigroup of "A" containing all elements except the empty string. It is usually denoted "A". - In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra. - In computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not. It generalizes the concept of a string, by not forcing the letters to always be in a fixed order, but allowing certain reshufflings to take place. Traces were introduced by Cartier and Foata in 1969 to give a combinatorial proof of MacMahon's Master theorem. Traces are used in theories of concurrent computation, where commuting letters stand for portions of a job that can execute independently of one another, while non-commuting letters stand for locks, synchronization points or thread joins. - Multiplication (often denoted by the cross symbol "", by a point "", by juxtaposition, or, on computers, by an asterisk "") is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division. - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - In mathematics , an invertible element or a unit in a ( unital ) ring R is any element u that has an inverse element in the multiplicative monoid of R , i.e. an element v such that uv = vu = 1R , where 1R is the multiplicative identity . The set of units of any ring is closed under multiplication ( the product of two units is again a unit ) , and forms a group for this operation . It never contains the element 0 ( except in the case of the zero ring ) , and is therefore not closed under addition ; its complement however might be a group under addition , which happens if and only if the ring is a local ring . The term unit is also used to refer to the identity element 1R of the ring , in expressions like ring with a unit or unit ring , and also e.g. ' unit ' matrix . For this reason , some authors call 1R '' unity '' or '' identity '' , and say that R is a '' ring with unity '' or a '' ring with identity '' rather than a '' ring with a unit '' . The multiplicative identity 1R and its opposite 1R are always units . Hence , pairs of additive inverse elements x and x are always associated . - Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. - In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. The binary operation of a semigroup is most often denoted multiplicatively: "x""y", or simply "xy", denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all "x", "y" and "z" in the semigroup. - Concurrent computing is a form of computing in which several computations are executed during overlapping time periods"concurrently"instead of "sequentially" (one completing before the next starts). This is a property of a systemthis may be an individual program, a computer, or a networkand there is a separate execution point or "thread of control" for each computation ("process"). A "concurrent system" is one where a computation can advance without waiting for all other computations to complete. - In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them. This concept is used in algebraic structures such as groups. The term "identity element" is often shortened to "identity" (as will be done in this article) when there is no possibility of confusion. - In mathematics and computer science, the syntactic monoid "M"("L") of a formal language "L" is the smallest monoid that recognizes the language "L". - A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the essence of a term) and extensional definitions (which proceed by listing the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. - In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thread joins) for providing rendezvous points between a set of independently executing processes or threads. - In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set. More formally, a binary operation is an operation of arity two whose two domains and one codomain are the same set or subsets thereof. Examples include the familiar elementary arithmetic operations of addition, subtraction, multiplication and division. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups. - In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function. For instance, the functions and can be "composed" to yield a function which maps in to in . Intuitively, if is a function of , and is a function of , then is a function of . The resulting "composite" function is denoted , defined by for all in. The notation is read as " circle ", or " round ", or " composed with ", " after ", " following ", or " of ", or " on ". Intuitively, composing two functions is a chaining process in which the output of the inner function becomes the input of the outer function. - The star height problem in formal language theory is the question whether all regular languages can be expressed using regular expressions of limited star height, i.e. with a limited nesting depth of Kleene stars. Specifically, is a nesting depth of one always sufficient? If not, is there an algorithm to determine how many are required? The problem was raised by . - Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. A small quantity is sometimes referred to as a quantulum. - In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group. - Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together, making a total of five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" i.e., "3 "add" 2 is equal to 5". - In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. - In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity. Monoids occur in several branches of mathematics; for instance, they can be regarded as categories with a single object. Thus, they capture the idea of function composition within a set. In fact, all functions from a set into itself form naturally a monoid with respect to function composition. Monoids are also commonly used in computer science, both in its foundational aspects and in practical programming. The set of strings built from a given set of characters is a free monoid. The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history monoids provide a foundation for process calculi and concurrent computing. Some of the more important results in the study of monoids are the KrohnRhodes theorem and the star height problem. The history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups. After reading the paragraphs above, we are interested in knowing the entity with which 'unit ' exhibits the relationship of 'instance of'. Find the answer from the choices below. Choices: - addition - algebra - algebraic structure - application - article - binary operation - branch - change - circle - collection - combination - communication - composition - computer - concept - cross - definition - design - division - five - formal language - function - history - idea - identity - identity element - information - language - limited - list - magnitude - mathematics - matrix - may - notation - number - object - operation - order - point - programming - property - quality - quantity - range - relationship - science - set - set of characters - single - space - star - statement - string - study - symbol - theorem - theory - three - time - two - understanding - unit of measurement - vector - word
A:
concept