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"In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. Overview The term semifield has two conflicting meanings, both of which include fields as a special case. * In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is a nonassociative division ring with multiplicative identity element. More precisely, it is a nonassociative ring whose nonzero elements form a loop under multiplication. In other words, a semifield is a set S with two operations + (addition) and · (multiplication), such that ** (S,+) is an abelian group, ** multiplication is distributive on both the left and right, ** there exists a multiplicative identity element, and ** division is always possible: for every a and every nonzero b in S, there exist unique x and y in S for which b·x = a and y·b = a. : Note in particular that the multiplication is not assumed to be commutative or associative. A semifield that is associative is a division ring, and one that is both associative and commutative is a field. A semifield by this definition is a special case of a quasifield. If S is finite, the last axiom in the definition above can be replaced with the assumption that there are no zero divisors, so that a*b = 0 implies that a = 0 or b = 0. Note that due to the lack of associativity, the last axiom is not equivalent to the assumption that every nonzero element has a multiplicative inverse, as is usually found in definitions of fields and division rings. * In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all nonzero elements have a multiplicative inverse. These objects are also called proper semifields. A variation of this definition arises if S contains an absorbing zero that is different from the multiplicative unit e, it is required that the non-zero elements be invertible, and a·0 = 0·a = 0. Since multiplication is associative, the (non-zero) elements of a semifield form a group. However, the pair (S,+) is only a semigroup, i.e. additive inverse need not exist, or, colloquially, 'there is no subtraction'. Sometimes, it is not assumed that the multiplication is associative. Primitivity of semifields A semifield D is called right (resp. left) primitive if it has an element w such that the set of nonzero elements of D* is equal to the set of all right (resp. left) principal powers of w. Examples We only give examples of semifields in the second sense, i.e. additive semigroups with distributive multiplication. Moreover, addition is commutative and multiplication is associative in our examples. * Positive rational numbers with the usual addition and multiplication form a commutative semifield. *:This can be extended by an absorbing 0. * Positive real numbers with the usual addition and multiplication form a commutative semifield. *:This can be extended by an absorbing 0, forming the probability semiring, which is isomorphic to the log semiring. * Rational functions of the form f /g, where f and g are polynomials in one variable with positive coefficients, form a commutative semifield. *:This can be extended to include 0. * The real numbers R can be viewed a semifield where the sum of two elements is defined to be their maximum and the product to be their ordinary sum; this semifield is more compactly denoted (R, max, +). Similarly (R, min, +) is a semifield. These are called the tropical semiring. *:This can be extended by −∞ (an absorbing 0); this is the limit (tropicalization) of the log semiring as the base goes to infinity. * Generalizing the previous example, if (A,·,≤) is a lattice-ordered group then (A,+,·) is an additively idempotent semifield with the semifield sum defined to be the supremum of two elements. Conversely, any additively idempotent semifield (A,+,·) defines a lattice-ordered group (A,·,≤), where a≤b if and only if a + b = b. * The boolean semifield B = {0, 1} with addition defined by logical or, and multiplication defined by logical and. See also * Planar ternary ring (first sense) References Category:Algebraic structures Category:Ring theory "
"Richard Michael Pyrah (born 1 November 1982, Dewsbury, Yorkshire, England), known as 'Rich', is an English first-class cricketer, who played all his career for Yorkshire County Cricket Club. Educated at Ossett School, the right-hand batsman and right-arm medium pacer was used mainly in one day and Twenty20 cricket. He made his one day debut in 2001, but had to wait until 2004 for his first-class bow. Pyrah made his 100th appearance for Yorkshire in one day cricket in June 2013, against Middlesex at Headingley. He is married to Lucie and they have twin daughters, Millie and Tilly. In September 2015, at the end of his benefit season, Pyrah retired from professional cricket in order to take up a full-time post on the club's coaching staff. ReferencesExternal links *Cricinfo Profile *Cricket Archive Statistics Category:1982 births Category:Living people Category:Yorkshire cricketers Category:Cricketers from Dewsbury Category:English cricketers of the 21st century Category:English cricketers Category:Yorkshire Cricket Board cricketers Category:English cricket coaches "
"is a Japanese word meaning "unevenness; irregularity; lack of uniformity; nonuniformity; inequality",Kenkyusha's New Japanese-English Dictionary (2003), 5th edition, Tokyo: Kenkyusha, p. 2536. and is a key concept in the Toyota Production System (TPS) as one of the three types of waste (muda, mura, muri). Waste reduction is an effective way to increase profitability. Toyota adopted these three Japanese words as part of their product improvement program, due to their familiarity in common usage. Mura, in terms of business/process improvement, is avoided through Just-In-Time systems which are based on keeping little or no inventory. These systems supply the production process with the right part, at the right time, in the right amount, using first-in, first-out (FIFO) component flow. Just-In-Time systems create a “pull system” in which each sub-process withdraws its needs from the preceding sub- processes, and ultimately from an outside supplier. When a preceding process does not receive a request or withdrawal it does not make more parts. This type of system is designed to maximize productivity by minimizing storage overhead. For example: # The assembly line “makes a request to,” or “pulls from” the Paint Shop, which pulls from Body Weld. # The Body Weld shop pulls from Stamping. # At the same time, requests are going out to suppliers for specific parts, for the vehicles that have been ordered by customers. # Small buffers accommodate minor fluctuations, yet allow continuous flow. If parts or material defects are found in one process, the Just-in-Time approach requires that the problem be quickly identified and corrected. Implementation Production leveling, also called heijunka, and frequent deliveries to customer are key to identifying and eliminating Mura. The use of different types of Kanban to control inventory at different stages in the process are key to ensuring that "pull" is happening between sub-processes. Leveling production, even when different products are produced in the same system, will aid in scheduling work in a standard way that encourages lower costs. It is also possible to smooth the workflow by having one operator work across several machines in a process rather than having different operators; in a sense merging several sub-processes under one operator. The fact that there is one operator will force a smoothness across the operations because the workpiece flows with the operator. There is no reason why the several operators cannot all work across these several machines following each other and carrying their workpiece with them. This multiple machine handling is called "multi-process handling" in the Toyota Production System. Limitations, critiques and improvements Some processes have considerable lead time. Some processes have unusually high costs for waiting or downtime. When this is the case, it is often desirable to try to predict the upcoming demand from a sub-process before pull occurs or a card is generated. The smoother the process, the more accurately this can be done from analysis of previous historical experience. Some processes have asymmetric cost. In such situations, it may be better to err away from the higher cost error. In this case, there appears to be waste and higher average error, but the waste or errors are smaller ones and in aggregate leads to lower costs / more customer value. For example, consider running a call center. It may be more effective to have low cost call center operators wait for high value clients rather than risk losing high value clients by making them wait. Given the asymmetric cost of these errors - particularly if the processes are not smooth - it may be prudent to have what seems like a surplus of call center operators that appear to be "wasting" call center operator time, rather than commit the higher-cost error of losing the occasional high value client. See also * Fukushima Daiichi nuclear disaster, this disaster is said to be caused by "nuclear mura". References Category:Japanese business terms Category:Lean manufacturing "