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...rd rich, vivid lives. They worked for those parents and mentors. They can and will work for you too. Mantras are the 21st Century way to lock your ideals, standards, ethics, and principles into formative minds. By definition they demand repetition. The phrasing may stay the same or almost the same. The stories, the elaboration, the background, the colors may bob and wea...

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...sophic interval bisemigroups or neutrosophic biinterval semigroups. We derive results pertaining to them. The new notion of quasi bisubsemigroups and ideals are introduced. Smarandache interval neutrosophic bisemigroups are also introduced and analysed. Also notions like neutrosophic interval bigroups and their substructures are studied in section two of this chapter. Neut...

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...oward rich, vivid lives. They worked for those parents and mentors. They can and will work for you too. Mantras are the 21st Century way to lock your ideals, standards, ethics, and principles into formative minds. By definition they demand repetition. The phrasing may stay the same or almost the same. The stories, the elaboration, the background, the colors may bob and wea...

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............... 75 Parenting Techniques .............................................................................................. 78 1—Build your ideals and values into your children. ........................................ 78 2—Build empathy with ordinary people. ......................................................... 79 DAVID R. WOMMACK 5 3—Firmly reject racia...

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...d to give up the ideal of perfection and the ideal of sanctity, which were ideals I had cultivated with complete devotion for more than twenty years....

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...2 2 The picture on the cover is the lattice representation of the S-ideals of the Smarandache mixed direct product ring R = Z 3 × Z 12 × Z 7... ...ring and a Smarandache ring. For, in a ring the lattice representation of ideals is always a modular lattice but we see in case of S-rings the lattic... ...l II. It is important to mention that unlike in rings where the two sided ideals form a modular lattice, we see in case of Smarandache rings the two... ...form a modular lattice, we see in case of Smarandache rings the two sided ideals in general do not form a modular lattice which is described in the ... ...ngs. Several types of semigroups are defined and their substructures like ideals and subsemigroups are also defined and illustrated with several exa... ...ind all subsemigroups of the symmetric semigroup S(6). 7. Find all right ideals of the symmetric semigroup S(9). 8. Find only ideals of the symmet... ... × 3 matrices with entries from Z 2 = {0,1}. Is S 3×3 a semigroup? Find ideals and subsemigroups in S 3×3 . Does S 3×3 have idempotents? Does S 3... ... 11} under multiplication modulo 12. Find i. Subsemigroups which are not ideals. ii. Ideals. iii. Zero divisors. iv. Idempotents. v. Units. ... ... Idempotents. v. Units. 12. Find in the semigroup S(21) right and left ideals. Does S(21) have subsemigroups which are not ideals? 1.3 Latt...

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... an ideal of R. DEFINITION 2.2.18: Let R be a SNA-ring if R has no SNA-ideals then we say R is SNA-simple ring. We define SNA-quotient ring f... ...this question is little complicated as we have the very definition of SNA-ideals only relative to SNA- subrings and not relative to the whole ring R.... ... the whole ring R. So one may presume that the quotient ring may have SNA-ideals relative to some other SNA-subrings. Thus when we say the ring R... ...us when we say the ring R is SNA-simple it is assumed the ring has no SNA- ideals related to any of the SNA-subrings. One may be wanting to know the c... ...may be wanting to know the concept of Smarandache maximal and SNA-minimal ideals. For the sake of helping the reader we define explicitly these type... ...s. For the sake of helping the reader we define explicitly these types of ideals. DEFINITION 2.2.19: Let R be a ring. S a SNA-subring of R. We sa... ...then L = (0) or L = J. 29 Unlike in the case of minimal and maximal ideals we see in case of SNA-ideals the maximality or the minimality is on... ... 4 3 5 2 e 1 5 5 2 4 1 3 e Find a. SNA-subrings of Z 3 L. b. SNA-ideals if any in Z 3 L. c. SNA-principal ideal if any. d. SNA-maximal / ... ...n in problem (6) and Z the ring of integers SNA-subrings which has no SNA-ideals related to them. 8. Does there exists for loop ring Z 7 L 13 (5...

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...nder multiplication. Clearly I = {0, 7} and J = {0, 2, 4, 6, 8, 10, 12} are ideals of Z 14. Example 1.1.9: Let X = {1, 2, 3, … , n} where n is a... ... 3 described in example 1.1.9.) It is left for the reader to find two sided ideals of S(3). 9 Notation: Throughout this book S(n) will denote t... ...tiplication. i. Find the number of elements in S 3 ×3 . ii. Find all the ideals of S 3 ×3 . iii. Find only the right ideals of S 3 ×3 . iv. Find... ...composition of mappings. i. Find all subsemigroups of S(X) which are not ideals. ii. Find all left ideals of S(X). iii. How many two sided ideal... ... S(X). iii. How many two sided ideals does S(X) contain? 3. Find all the ideals of Z 28 = {0, 1, 2, …, 27}, the semigroup under multiplication m... ...S(4) and Z 256 ? Justify your answer with reasons. 6. Find all the right ideals of S(5). Can S(5) have ideals of order 120? 7. Find all the sub... ...aneously both a right and a left ideal of R we say I is an ideal of R. Thus ideals are subrings but all subrings are not ideals. Example 1.3.14: L... ...o prove R / I is a ring. PROBLEMS: 1. Let F be a field. Prove F has no ideals. 2. Find all ideals of the ring Z 24 . 3. Prove Z 29 has no... ...be a field? 22 9. Find [] > + < 1 x x Z 2 . 10. Find all principal ideals in [] x Z 5 3 = {all polynomials of degree ≤ less than or equal ...

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...orking on concepts like fuzzy semigroup, fuzzy groups, fuzzy rings, fuzzy ideals, fuzzy semirings, fuzzy near-rings and so on. In this book, we ... ...ified. In 1982 [73] defined and studied fuzzy subrings as well as fuzzy ideals. Subsequently among [148, 149] fuzzified certain standard concepts ... ...quently among [148, 149] fuzzified certain standard concepts on rings and ideals. Here we just recall some of the results on fuzzy rings and leave i... ...a fuzzy ideal of R if and only if the level subsets µ t , t ∈ Im(µ ) are ideals of R. Proof: Left for the reader as it can be proved using the d... ...ion. THEOREM 1.4.3: If µ is any fuzzy ideal of a ring R, then two level ideals 2 1 t t and µ µ (with t 1 < t 2 ) are equal if and only if there i... ...ve as a matter of routine. This theorem gives an insight that the level ideals of a fuzzy ideal need not be distinct. 41 THEOREM 1.4.4: The l... ...s of a fuzzy ideal need not be distinct. 41 THEOREM 1.4.4: The level ideals of a fuzzy ideal µ form a chain. That is if Im µ = {t 0 , t 1 , …, ... ... µ = {t 0 , t 1 , …, t n } with t 0 > … > t n , then the chain of level ideals of µ is given by R n 1 0 t t t = ⊂ ⊂ ⊂ µ µ µ K . Proof: Straigh... ...constant fuzzy ideal µ of a ring R is called fuzzy prime if for any fuzzy ideals µ 1 and µ 2 of R the condition µ 1 µ 2 ⊆ µ implies that either µ...

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...Z 6 = {0, 1, 2, 3, 4, 5}. Clearly I = {0, 3} and J = {0, 2, 4} are not S-ideals only ideals. Example 1.4.5: Let Z 12 = {0, 1, 2, ... , 11} be th... ...lled a Smarandache ideal II (S-ideal II) of R related to A. If R has no S-ideals I (II) we call R a Smarandache simple ring (S-simple ring) I (II), ... ... chain conditions (S-A.C.C for brevity) if for every ascending chain of S- ideals I j of R that is I 1 ⊂ I 2 ⊂ I 3 ⊂ … is stationary in the sense ... ...C) for brevity) if every descending chain N 1 ⊃ N 2 ⊃ … ⊃ N k ⊃ … of S-ideals N j of R is stationary. On similar lines one can define Smarandache... ...r lines one can define Smarandache A.C.C and Smarandache D.C.C of S-right ideals and S-left ideals of a ring. Thus we say a ring R is said to be Sma... ... Smarandache left Noetherian (S-left Noetherian) if the S-A.C.C on S-left ideals is satisfied. Similarly we say a ring R is Smarandache right Artinia... ...y a ring R is Smarandache right Artinian (S-right Artinian) if the S-left ideals of R satisfies the S-D.C.C. condition. DEFINITION 1.4.30: Let R b... ...S-ideal I (or II) of R is generated by regular elements of I i.e. these S-ideals does not contain zero divisors. DEFINITION 1.4.40: Let R be a rin... ...marandache n-ideal ring (S-n-ideal ring) if for every set of n-distinct S-ideals I (or II), I 1 , I 2 , …, I n of R and for every distinct set of n...

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...multiplication mod 10. Clearly 5 2 ≡ 5(mod 10). PROBLEM 1: Find all the ideals in the semigroup Z 36 = {0, 1, .... , 35} under multiplication.... ...} under multiplication. PROBLEM 2: Find all the subsemigroups which are not ideals in R 2×2 given in example 1.3.4. PROBLEM 3: Find all the ide... ...he semigroup S 3×3 . S 3×3 given in example 1.3.6. PROBLEM 4: Find all the ideals and subsemigroups of Z 128 = {0, 1, 2, ... , 127}, Z 128 is a ... ... properties on subsets of a groupoid like subgroupoid, conjugate subgroupoid, ideals etc. and derive some interesting results using them. DEFINITI... ... a 3 a 1 a 3 a 1 a 3 P = {a 0 , a 2 } and Q = {a 1 , a 3 } are right ideals of G. Clearly P and Q are not left ideals of G. In fact G has n... ... ideals of G. Clearly P and Q are not left ideals of G. In fact G has no left ideals only 2 right ideals. 23 Example 2.2.4: Let G be a groupo... ...3 Clearly, G has P = {a 0 , a 2 } and Q = {a 1 , a 3 } to be the only left ideals of G and G has no right ideals. Now we proceed to define normal... ... H and K are conjugate with each other. To get more normal subgroupoids and ideals, that is to make groupoids have richer structures we define dir... ... for the reader to find the subgroupoids, if any, normal subgroupoids if any, ideals if any (right, left) and so on. PROBLEM 1: Can a groupoid o...

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...in the future that it establishes its activities on the interests and works for the ideals of struggling Macedonia and the Bulgarian race." TODOR A... ...r harbours no sympathy towards the Ottoman Empire. The IMRO was fighting for lofty ideals (Balkanian federation) and worthy goals (liberation from ... ...nd the biggest and weightiest amongst them by far. It is founded on the vision and ideals of Goce Delcev and has distanced itself from the "Terrori... ...onne was never a good preparatory school to the academy of Balkan reality. Rugova's ideals were good and noble - Gandhi-like passive resistance, mar... ...ery if not to outright criminality. It is a fine balancing act between the virtuous ideals of democracy and the harsh constraints of realpolitik. ... ...ed brilliance, bodily beauty or performance, or ideal, everlasting, all-conquering ideals or political theories. 3. The group as a whole, or memb...

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...r of someone else, and to finally dedicate ourselves to “our own” needs and ideals. 8 We can say the same thing when we look at the experience of ... ...at be true Joy? COMMENT: Couldn’t Joy also be what those who reach their ideals experience, like martyrs? ANTONIO: Sadomasochistic pleasure exi...

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...zl died in 1904, having actually been deposed by Zionists from Russia who espoused ideals of equality for all, Jews and non-Jews alike. The Holoca... ...he judiciary, and the world of business. This clubbish rendition of the democratic ideals is a travesty and a mockery. People power is the inevitab... ...apoleon justified his gory campaigns by claiming that they served to spread French ideals throughout a barbarous world. Kipling bemoaned the "White... ...eating a new communal or reciprocal productive relationship through new priorities, ideals, and extensive governmental control and regulation. The g... ...ncial turmoil engendered by the Clinton affair, would you still stick to your lofty ideals?" AC: "You inadvertently, I am sure, broached the heart o... ...ntuition? Perhaps intuition is the mind's way of interacting directly with Platonic ideals or Phenomenological "essences"? By "directly" I mean wit... ...thoughts and feelings that precede any intellectual analysis and underlie it. Moral ideals and rules may be such intuitions (see "Morality - a State... ...ion that a fracturing of the solutions occurs. Atrocities are the vile daughters of ideals. Armed with stereotypes - those narcissistic defence mech... ...ed brilliance, bodily beauty or performance, or ideal, everlasting, all-conquering ideals or political theories. 3. The group as a whole, or memb...

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...Clifford aggregate. Also, spinors are the members of left or right minimal ideals of Clifford algebra, which may provide the framework for a deeper un... ...ensionality. As we have already indicated, since spinors are in left/right ideals of a Clifford algebra, a supersymmetry is then naturally incorporate... ...present paper. Ordinary spinors are nothing but elements of the left/right ideals of a Clifford algebra. So they are automatically con- tained in the ...

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...s throughout history, the overall patriarchal nature of society and social ideals has been steadily increasing in strength, partly because of a tenden... ...of reciprocity rather than domination and submission.” (p. 10) The overall ideals of the American Indians were conserva- tional, in contrast to the Ju... ...hat is, the as- pects of your personality that don’t meet approved ethical ideals 150. Wallace, 1970, p. 30; also quoted in Stites, 1904. THE RELIGIOU... ...ELIGIOUS DIMENSION 230 toward family and community. The Iroquois cultural ideals called for love and brotherhood in these contexts. In assuming respo...

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...evils require extreme remedies, if desperate conditions require impossible ideals, why not decide to not fall into the common pit? Only by proposin... ...ows of the dead are the traces left behind from the passions and the false ideals that human beings allow to govern their lives. Achilles who cries... ...e who transcend their reactions and live following values and virtue, with ideals and projects to build upon. The reactive being transforms itself...

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...lisations. Inevitably, the inexorable processes and conflicts of life grind these ideals into the fine dust of the real. Disappointments follow di...

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...ink with self-respect! Not only post and positions, but also think about the ideals! We should live without disgrace. The only medicine for this is... ... all, now-citizens of one country. We have to work together for achieving our ideals as we are all classified under one caste. So far as I am concer... ...se of the public roads! Even though temple entry is not at present one of the ideals of Congress, so far as I am concerned it is one of my main ide...

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...eer formation, who is concerned with the apparent dilemma of reconciling altruistic ideals with current negative myths about the business world; (2) ... ...mediated by a hypothetical new ethic of consumption (Brown 1978) stressing loftier ideals than those available only from material goods. For example...

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...ephone handset. Carter petitioned against the rule and won. 5 Mindful of the ideals behind the Hush-A-Phone decision, the FCC agreed that so long as t... ...ree, draws upon their mean- ings. The Free Software Philosophy The normative ideals of the free software movement and the descriptive attri- butes of ...

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...rchiving. In this sense the BOAI is not just a statement of principles or ideals, but a serious and effective plan to achieve this very important pu... ... to the 35th Anniversary of Project Gutenberg. If you have any particular ideals or ideas you would like to have included in these events surroundin...

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...ictions" and "clubs" deal with this problem). Nonrivalry and nonexcludability are ideals - not realities. They apply strictly only to the sunlight...

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...to be above the laws they enforced. Only after he was gone, did the original ideals of the Revolution slowly begin to re- surface as mere token idea... ... butcher than Cyrus the Great. The Persian Golden Age of Empire… its higher ideals, its higher columns, its more beautiful art, its greater achiev...

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...ing: Have Separate Research-Intensive Universities and Teaching-Intensive Universities Is Academic Research Grant Money Corrupting Academic Teaching Ideals? CS & IT Academia: Is Teaching Excellence Important? Is a PhD in CS/IT Necessarily a Good Teacher? A Defense of The CS/IT PhD Teacher Suggestion of Separate University Rankings for Research Excellence and Teaching ...