wiktionary ternaryTernary from Latin ternarius is an adjective meaning composed of three items . It can refer to Ternary complex , a complex formed by the interaction of three molecules Ternary compound , a type of chemical compound Ternary computer , a computer using a ternary numeral system Ternary form , a form used for structuring music Ternary Golay code , a perfect 11, 6, 5 ternary linear code Ternary heap , a data structure in computer science Ternary logic , a logic system with the values true , false , and some other value Ternary name for any taxon below the rank of species Ternary numeral system , a base 3 counting system Balanced ternary , a positional numeral system, useful for comparison logic Ternary operation , an operation that takes three parameters Ternary plot or ternary graph, a plot that shows the ratios of three proportions Ternary relation , a finitary relation in which the number of places in the relation is three Ternary ring operator , a linear or vector space M of operators between two Hilbert spaces such that M M M M Ternary search , a computer science technique for finding the minimum or maximum of a function Ternary signal , a signal that can assume three significant values Ternary tree , a tree data structure in computer science Ternary search tree , a ternary three way tree data structure of strings See also Unary disambiguation 1 Binary disambiguation 2 Tertiary disambiguation 3rd Quaternary disambiguation 4 Quinary 5 Senary 6 Septenary 7 disambig Category Latin adjectives in current use de Tern r nl Ternair ... more details
Image Min T Heap.png thumb right 180px A diagram of a min ternary heap. Image Max T Heap.png thumb right 180px A diagram of a max ternary heap. A ternary heap is a data structure in computer science . It is part of the Heap data structure heap family. It inherits the name from ternary tree and the Heap data structure heap . Often there are two kinds of ternary heaps the min ternary heap and the max ternary heap . Min Ternary Heap As illustrated, a min ternary heap has the smallest element as its root. Parent node has to be smaller than or equal to its children. Max Ternary Heap A max ternary heap has the biggest element as its root. Each node has to be greater than or equal to its children. In both cases, a ternary heap has to satisfy the heap property, which is every level of tree has to be filled, or if the last level is not filled, the leaves are distributed from left to right. These rules has to be strictly followed when implementing the code to build a ternary heap. Heapify The process of building a heap is called Heapify . A way to implement this data structure is to use a Collection computing collection , e.g. Vectors in Java. Each element in the collection is a node of the ternary heap, where the index is in the order of from top to bottom, from left to right. The example max ternary heap in a Vector will be like this pre 99, 63, 56, 88, 10, 53, 42, 36, 18, 39 pre For each node, assuming that its index is i, and that indexing starts with node 0 parent s index math floor i 1 3 math left child s math 3 i 1 math middle child s math 3 i 2 math right child s math 3 i 3 math The following C like pseudocode shows how to heapify a vector to a max ternary heap. ref Cite book last Cormen first Thomas authorlink coauthors Charles E. Leiserson title Introduction to Algorithms publisher date location pages 128 130 url doi id isbn ref source lang C Max Heapify A,i largest 1 ... A, largest source See also Binary heap Binary Heap Ternary tree Ternary Tree references Category Data ... more details
A ternary computer also trinary computer is a computer that uses ternary logic three possible values ..., was a ternary computer. The only modern, electronic ternary computer Setun was built in the late ... of the computer, which he called Setun 70. Balanced ternary Main Balanced ternaryTernary computing is commonly implemented in terms of balanced ternary , which uses the three digits 1, 0, and 1. The negative value of any balanced ternary digit can be obtained by replacing every with a and vice versa ... ternary can express negative values as easily as positive ones, without the need for a leading negative sign as with decimal numbers. These advantages make some calculations more efficient in ternary than binary. I often reflect that had the Ternary instead of the binary Notation been adopted in the Infancy ... for computers, ternary computers have diminished to a small footnote in the history of computing. However, ternary logic s elegance and efficiency is predicted by Donald Knuth to bring them back ... this can happen is by the combination of an optical computer with the ternary logic system. ref http www.iop.org EJ abstract 1402 4896 2005 T118 025 Ternary Optical Computer ref A ternary computer ... potential has also been remarked by certain companies as Hypres which is actively engaged in ternary computing. IBM also reports infrequently on ternary computing topics in its papers , but it is not actively engaged in it. The Josephson junction has been proposed as a balanced ternary memory ... and very simple construction with less number of elements due to the ternary operation. ref doi 10.1109 ISMVL.1998.679270 ref In 2009, a ternary quantum computer was proposed which thus uses qutrit ... . ref http nextbigfuture.com 2009 08 qudits multilevel versions of qubits.html Qudits ref Ternary ... ternary system. Minerva, in reporting a calculation result, says three hundred forty one thousand ... Ternary Computer. Institut f r Buchkunst Leipzig, 2008, ISBN 3 932865 48 0 English, German See also ... more details
Unreferenced auto yes date December 2009 File Trisodium phosphate.png right thumb 100px Sodium phosphate , Na sub 3 sub PO sub 4 sub , is a ternary compound. In chemistry , a ternary compound is a compound containing three different elements. An example of this is sodium phosphate , Na sub 3 sub PO sub 4 sub . The sodium ion has a charge of 1 and the phosphate ion has a charge of 3 . Therefore, three sodium ions are needed to balance the charge of one phosphate ion. Another example of a ternary compound is calcium carbonate CaCO sub 3 sub . See also Binary compound DEFAULTSORT Ternary Compound Chemistry stub Category Chemical compounds ... more details
Table Numeral Systems Balanced ternary is a Non standard positional numeral systems non standard positional numeral system a balanced form , useful for comparison logic. It is a Ternary numeral system ternary system, but unlike the standard unbalanced ternary system, the digits have the values 1, 0, and 1 ... possible relationships are less than, equals, and greater than. Balanced ternary can represent all integers without resorting to a separate minus sign . Balanced ternary is counted as follows ... f9f9f9 color black border 1px aaa solid border collapse collapse Balanced ternary align center Decimal 6 5 4 3 2 1 0 1 2 3 4 5 6 align center Balanced ternary u 1 u 10 u 1 u 11 u 11 u u 1 u 0 u 1 u 1 u 1 u 0 1 1 u 1 u 10 11 1 u 11 u 1 u 1 u 0 Unbalanced ternary can be converted to balanced ternary ... u 1 u 1 sub 3 bal sub 7 sub 10 sub . Balanced ternary is easily represented as electronic signals, as potential ... ternary has other applications besides computing. For example, a classical 2 pan Weighing scale Balance ... 2011 Similarly, a currency system using ternary values would save visits to the bank customers would ... days of computing that were built with balanced ternary instead of binary. The notation has a number of computational advantages over regular binary, and over traditional ternary. Particularly, the one digit multiplication table has no carries in balanced ternary, and the addition table has only two ... sum is zero. Fractional balanced ternary Balanced ternary can be extended to fractional numbers similarly ... and rounding are the same operation in balanced ternary they produce exactly the same ... span style text decoration overline 9 span sub 10 sub , in balanced ternary such numbers have only ..., multiplication, and division&mdash are done as in regular ternary, except that there are some special considerations on the size compare. Multiply 2 by 10 in ternary and then divide the result ... division . As in binary, there are balanced ternary equivalents of shift and add multiplication ... more details
In telecommunication , a ternary signal is a Signalling telecommunication signal that can assume, at any given instant, one of three states or significant condition s, such as Power physics power level, phase waves phase position, pulse duration , or frequency . Examples of ternary signals are a a pulse that can have a positive, zero, or negative voltage value at any given instant pulse amplitude modulation PAM 3 , b a sine wave that can assume phases of 0 , 120 , or 240 relative to a clock pulse 3 phase shift keying PSK , and c a carrier wave that can assume any one of three different frequencies depending on three different modulation signal significant conditions 3 frequency modulation FM . Modulation techniques Some examples of PAM 3 line code s that use ternary signals are hybrid ternary code bipolar encoding MLT 3 encoding used in Fast Ethernet 100BASE TX 100BASE TX Ethernet B3ZS 4B3T used in some ISDN basic rate interface 8B6T used in Fast Ethernet 100BASE T4 100BASE T4 Ethernet return to zero DQPSK SOQPSK SOQPSK TG uses ternary continuous phase modulation https dspace.byu.edu handle 1877 58 3 PSK can be seen as falling between binary phase shift keying BPSK which uses two phases, and quadrature phase shift keying QPSK which uses four phases. References FS1037C See also balanced ternary digital signal Fast Ethernet 100BASE T2 uses PAM 5, which, like ternary, is one of the few modulation schemes that does not use a power of two number of symbols. three phase electric power , like 3 PSK, uses 3 phases at a single frequency and amplitude. DEFAULTSORT Ternary Signal Category Line codes Telecomm stub de Tern res Signal ... more details
unreferenced date October 2010 Ternary form is a three part A B A structure in a piece of music. The first and third parts A are musically identical, or very nearly so, while the second part B contrasts sharply with it. The B section is often known as a trio music trio . The first section of a piece in ternary form does not usually change key music key . The middle section will generally be in a different key, often the subdominant of the first section a perfect fifth below or the Relative key relative minor or major. It usually also has a contrasting character in a march music march , for example, the highly rhythmic and strident character of the march itself is usually contrasted with a more lyrical and flowing trio. Often the trio is in a Triple metre 3 4 time signature as opposed to the 4 4 of the primary march theme. Commonly, the third section will feature more ornamentation than the first section e.g. da capo aria s . In these cases the last section is sometimes labeled A or A1 to indicate that it is slightly different from the first A section. As well as in marches, ternary form is often found in baroque music baroque opera aria s the da capo aria and in many dance forms, such as polka s. It is also the form used in the minuet or scherzo and trio, which in the Classical period music classical music era was usually the third movement of symphony symphonies , string quartet s, sonata s and similar works. A distinction is sometimes made between compound ternary form in which each large part of the form is itself divided in a way to suggest ternary or binary form giving, for example, an overall scheme of Nowrap A B A C D C A B A and simple ternary form , in which each large part of the form has no particular structure itself. Da capo arias are usually in simple ternary form minuets or scherzos and trios are normally compound. Another name for the latter is composite ternary form. Category Musical form musical form music theory stub es Forma ternaria ko ja ... more details
Unreferenced date December 2009 In mathematics , a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3 adic , 3 ary , 3 dimensional , or 3 place . Just as a binary relation is formally defined as a set of pairs , i.e. a subset of the Cartesian product Nowrap A B of some sets A and B , so a ternary relation is a set of triples, forming a subset of the Cartesian product Nowrap A B C of three sets A , B and C . An example of a ternary relation in elementary geometry is the line geometry Collinear points collinearity of points . Example graph of a binary function Further Graph of a function binary function A function Nowrap A B C in two variables, taking values in two sets A and B , respectively, is formally a function that associates to every pair a , b in Nowrap A B an element a , b in C . Therefore its graph consists of pairs of the form Nowrap a , b , a , b . Such pairs in which the first element is itself a pair are often identified with triples. This makes the graph of a ternary relation between A , B and C , consisting of all triples Nowrap a , b , a , b , for all a in A and b in B . Example cyclic orders Main Cyclic order Given any set A whose elements are arranged on a circle, one can define a ternary relation R on A , i.e. a subset of A sup 3 sup Nowrap A A A , by stipulating that Nowrap R a , b , c holds if and only if the elements a , b and c are pairwise different and when going from a to c in a clockwise direction one passes through b . For example if A Nowrap 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 represents the hours on a clock face , then Nowrap R 8, 12, 4 holds and Nowrap R 12, 8, 4 does not hold. DEFAULTSORT Ternary Relation Category Mathematical relations es Relaci n ternaria pt Rela o tern ria ru ... more details
For the ternary Conditional programming if operator in C programming language C like languages ? In mathematics , a ternary operation is an n ary operation with n 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A. An example of a ternary operation is the product in a heap mathematics heap . In computer science a ternary operator sometimes ... feature a ternary operator, code ? code , which defines a conditional programming If expressions conditional expression . Since this operator is often the only existing ternary operator in the language, it is sometimes simply referred to as the ternary operator , though it is more accurately referred ... be rewritten as the following ternary statement source lang c result a b ? x y source Common Lisp Assignment using a ternary expression in Common Lisp source lang lisp setf result if a b x y source Imperative ... to use the ternary operator source lang perl result a b ? x y source As of version 5.3, PHP supports a ternary shortcut syntax tt expr1 ? expr2 tt which is equivalent to tt expr1 ? expr1 expr2 tt ... lang PHP result condition ? ifFalse source Nested ternary operators are allowed in PHP Perl but are not always ... web accessdate 2010 04 19 url http www.kwista.com programming ternary operators title Ternary Operators ... the nested ternary operator source lang PHP result a b ? a c ? a c b c ? b c source Perl 6 Uses double ... over syntax, a ternary operator for Python programming language Python was approved as http ... 2006. Python s ternary operator differs from the common code ? code operator in the order ... isn t consistent with other languages ternary operators, e.g. by using an explicitly constructed dictionary ... if a b x else y end source These can be rewritten as the following ternary statement source lang ruby result a b ? x y source Visual Basic Visual Basic .NET Visual Basic Version 9 has added a ternary ... the type of the expression based on the types of its operands. See also ? , the ternary conditional ... more details
Refimprove date January 2010 A Ternary complex refers to a protein complex containing three different molecules which are bound together. In structural biology ternary complex can be used to describe a crystal containing a protein with two small molecules bound, for example cofactor biochemistry cofactor and Substrate biochemistry substrate or a complex formed between two proteins and a single substrate. ref Cite book title Protein methyltransferases first1 Steven G. last1 Clarke first2 Fuyuhiko last2 Tamanoi pages 162 172 publisher Academic Press year 2006 isbn 9 7801 2122 7258 ref In Immunology , a ternary complex can refer to the MHC Peptide T cell receptor complex formed when T cells recognize epitopes of an antigen. A ternary complex can also refer to a polymer formed by electrostatic interactions. ref Cite book title Trends in colloid and interface science XVII first1 Val rie last1 Cabuil first2 Pierre last2 Levitz first3 Clande last3 Treiner publisher Springer page 45 year 2004 isbn 9 7835 4020 0734 ref References Reflist Chemistry stub Biochemistry stub Molecular biology stub cs Tern rn komplex Category Protein complexes ... more details
on the triangular plot A ternary plot is drawn as a triangle. Each base, or side, of the triangle ... ternary diagram, without any points plotted. image ternary.example.axis.1.jpg An example ternary diagram, showing increments along the first axis. image ternary.example.axis.2.jpg An example ternary diagram, showing increments along the second axis. image ternary.example.axis.3.jpg An example ternary diagram, showing increments along the third axis. image Ternary plot 1.png Empty diagram image Ternary plot 2 reverse axis .png Empty diagram alternative axis gallery Using Cartesian coordinates ... an equilateral ternary plot where math a 100 math is placed at math x,y 0,0 math and math b 100 ... Cite web last Vaughan first Will title Ternary plots url http wvaughan.org ternaryplots.html date ... Flammability diagram QFL diagram Commonscat Ternary plots Category Diagrams de Dreiecksdiagramm ... more details
A ternary search algorithm is a technique in computer science for finding the maxima and minima minimum or maximum of a unimodal function function mathematics function that is either increasing function strictly increasing and then strictly decreasing or vice versa . A ternary search determines either that the minimum or maximum cannot be in the first third of the domain or that it cannot be in the last third of the domain, then repeats on the remaining two thirds. A ternary search is an example of a divide and conquer algorithm see search algorithm . The function Assume we are looking for a maximum of f x and that we know the maximum lies somewhere between A and B . For the algorithm to be applicable, there must be some value x such that for all a , b with A a < b x , we have f a < f b , and for all a , b with x a < b B, we have f a > f b . Run Time Order T n T n 3 c log n Recursive algorithm for ternary search source lang python a array elements k element to be searched i 0 j size of array 1 int tsearch int a,int i,int j,int k int m1,m2 m1 i j i 3 m2 j j i 3 if i j if k a m1 element found exit else if k a m2 element found exit if k a m1 return tsearch a,i,m1 1,k else if k a m2 return tsearch a,m2 1,j,k else return tsearch a,m1 1,m2 1,k else element not found exit source See also Binary search can be used to search for where the derivative changes in sign Newton s method in optimization can be used to search for where the derivative is zero Golden section search similar to ternary search, useful if evaluating f takes most of the time per iteration Interpolation search Linear search References Unreferenced date May 2007 Category Search algorithms Category Mathematical optimization ru ... more details
including itself. Properties of Ternary trees Maximum number of nodes Let h be height of a ternary tree. Let M h be the maximum number of nodes in a ternary tree of height h class wikitable h M h ... 3k . Right Child is stored in TREE 3k 1 . Common Operations Insertion Nodes can be inserted into ternary trees in between three other nodes or added after an external node . In Ternary trees, a node ... for null, and take a branch. i b s o a e h n t n t s y e f r o t Above picture is a balanced ternary ... Jon Bentley and Bob Sedgewick 1998 , Dr. Dobb s Journal ref Examples of Ternary tree Ternary search tree Ternary heap Ternary Heap An infinite ternary tree containing all primitive Pythagorean triples ... more details
f. multicaulis subf. surculosa Engl. & Irmsch. the ternary name would be Saxifraga aizoon subf. surculosa Engl. & Irmsch. The zoological equivalent of a ternary name is a trinomial name or trinomen . DEFAULTSORT Ternary Name Category Botanical nomenclature uk ... more details
In computer science , a ternary search tree is a ternary tree where the nodes are in order with respect to the search keys. If the search keys are String computer science strings , each node stores a single character and searching for a string consists of a series of binary searches, one for each character in the string if the current character in the search string is less than the character at the current node, the search goes to the left child node, if it is greater , the search goes to the right child node, if it is equal , the search goes to the middle child node and proceeds to the next character in the search string. The figure below shows the ternary search tree for it , cup , up , ma and us i c t u u m p p a s Like a binary search tree , a ternary search tree may be balanced or unbalanced, depending on insertion order if the tree is not self balancing . Searching for a string of length k in a balanced ternary search tree with n strings requires at most O k log n comparisons. This is because each comparison either consumes one character of the search string or halves the number of strings in the search space. As with other Trie tries , each node in a ternary search tree represents a Prefix computer science prefix of the stored strings. All strings in the middle subtree of a node start with that prefix. See also Search tree Hash table Hashtable References http www.drdobbs.com database 184410528 Ternary Search Trees Jon Bentley and Bob Sedgewick External links http www.cs.princeton.edu rs strings Ternary Search Trees http algs4.cs.princeton.edu 52trie TST.java.html Ternary Search Tree in Java http search.cpan.org mrogaski Tree Ternary 0.03 Ternary.pm Tree Ternary Perl module http dasnar.sdf eu.org res ctst README.html Ternary Search Tree code http abc.se re code tst STL compliant Ternary Search Tree in C http ternary.sourceforge.net Ternary Search Tree in C https github.com kanwei algorithms tree Ternary Search Tree in Ruby CS Trees Category Trees structure Category ... more details
Table Numeral Systems Ternary or trinary is the Base mathematics base num 3 numeral system . Analogous to a bit , a ternary numerical digit digit is a trit tr inary dig it . One trit contains math log 2 3 math about 1.58496 bits of information. Although ternary most often refers to a system in which ... its name to the balanced ternary system, used in comparison logic and ternary computer s. Comparison to other radixes Compared to decimal and binary Representations of integer number s in ternary do ... 365 number 365 corresponds to binary 101101101 9 digits and to ternary 111112 6 digits . However, they are still ... see below for a compact way to codify ternary using nonary and septemvigesimal. class wikitable Numbers one to twenty seven in standard ternary align center Ternary 1 2 10 11 12 20 21 22 100 align center ... center Ternary 101 102 110 111 112 120 121 122 200 align center Binary 1010 1011 1100 1101 1110 1111 10000 10001 10010 align center Decimal 10 11 12 13 14 15 16 17 18 align center align center Ternary ... 11010 11011 align center Decimal 19 20 21 22 23 24 25 26 27 class wikitable Powers of three in ternary align center Ternary 1 10 100 1 000 10 000 align center Binary 1 11 1001 1 1011 101 0001 align ... 4 sup align center align center Ternary 100 000 1 000 000 10 000 000 100 000 000 1 000 000 000 align ... 8 sup 3 sup 9 sup As for rational number s, ternary offers a convenient way to represent one third as opposed ... in decimal but a major drawback is that, in turn, ternary does not offer a finite representation ... wikitable Fractions in ternary align center Ternary 0. font style text decoration overline 1 font 11111111111 ... center align center Ternary 0. font style text decoration overline 01 font 0101010101... 0.01 0. font ... of the digits in ternary as opposed to binary Whereas in binary, where the sum of all previous digit values before 2 sup n sup can be found using the formula 2 sup n sup 1, in ternary the following formula ..., which is 7. In ternary the fourth digit has a value of 27 and the sum of all previous ternary numbers ... more details
News release date February 2008 Unreferenced date February 2008 The hybrid H ternary line code operates on a hybrid principle that combines the binary non return to zero level NRZ L and the polar return to zero RZ codes and thus it is called H ternary. The H ternary code has three levels for signal representation these are positive , zero 0 , and negative . These three levels are represented by three states. The state of the line code could be in any one of these three states. A transition takes place to the next state as a result of a binary input 1 or 0 and the encoder s present output state. The encoding procedure is as follows Glass and Bastaki 2001, Glass et al. 2001 . class wikitable style text align center Input bit Prior output Output level rowspan 3 0 rowspan 2 0 rowspan 2 0 rowspan 3 1 0 rowspan 2 In general, the encoder outputs level for a binary 1 input and a level for a binary 0 input. However, if this would result in the same output level as the previous bit time, a 0 level is output instead. Initially, the encoder output present state is assumed at 0 level when the first bit arrives at the encoder input. The new line coding scheme violates the encoding rule of NRZ L when a sequence of 1s or 0s arrives and hence, it overcomes some of their deficiencies. During the violation period for a run of 1s or 0s, it operates on the same encoding rule of the polar RZ but with pulse occupancy of full period. It is evident that NRZ L and polar RZ codes have deficiencies compared to the proposed H ternary encoding scheme. NRZ L code lacks sufficient timing information when the binary signal remains at one level in of either 1 or 0. This has direct influence on synchronising ... digital signal. The H ternary code has also timing superiority compared to similar ternary codes. Other ternary line code such as alternate mark inversion AMI also lacks the timing information ... bipolar encoding MLT 3 encoding Manchester encoding B3ZS 4B3T Bit encoding DEFAULTSORT Hybrid Ternary ... more details
There are two closely related error correcting code s known as ternary Golay codes . The code generally known simply as the ternary Golay code is a perfect code perfect 11, 6, 5 ternary linear code the extended ternary Golay code is a 12, 6, 6 linear code obtained by adding a zero sum check digit to the 11, 6, 5 code. Properties Ternary Golay code The ternary Golay code consists of 3 sup 6 sup 729 codewords. Its parity check matrix is math left begin array ccccccccccc 1 & 1 & 1 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 0 1 & 1 & 2 & 1 & 0 & 2 & 0 & 1 & 0 & 0 & 0 1 & 2 & 1 & 0 & 1 & 2 & 0 & 0 & 1 & 0 & 0 1 & 2 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & 1 & 0 1 & 0 & 2 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 1 end array right . math Any two different codewords differ in at least 5 positions. Every ternary word of length 11 has a Hamming distance of at most 2 from exactly one codeword. The code can also be constructed as the quadratic residue code of length 11 over the finite field F sub 3 sub . Used in a football pool with 11 games, the ternary Golay code corresponds to 729 bets and guarantees exactly one bet with at most 2 wrong outcomes. Extended ternary Golay code The enumerator polynomial complete weight enumerator of the extended ternary Golay code is math x 12 y 12 z 12 22 left x 6y 6 y 6z 6 z 6x 6 right 220 left x 6y 3z 3 y 6z 3x 3 z 6x 3y 3 right . math The automorphism group of the extended ternary Golay code is 2. M sub 12 sub , where M sub 12 sub is a Mathieu group . The extended ternary Golay code can be constructed as the span of the rows of a Hadamard matrix of order 12 over the field F sub 3 sub . Consider all codewords of the extended code which have just six nonzero digits. The sets of positions at which these nonzero digits occur form the Steiner system S 5, 6, 12 . History The ternary Golay code has been constructed by Marcel J. E. Golay . He published it in a very short correspondence in 1949. Independently and slightly earlier the same code had also been developed by the Finland ... more details
In mathematics , a ternary ring is an algebraic structure math R,T math consisting of a non empty set math R math and a ternary mapping math T colon R 3 to R math , and a planar ternary ring PTR or ternary field is special sort of a ternary ring used by harvtxt Hall 1943 to give coordinates to projective plane s. A planar ternary ring is not a Ring mathematics ring in the traditional sense. Definition A planar ternary ring is a structure math R,T math where math R math is a nonempty set, containing distinct elements called 0 and 1, and math T colon R 3 to R math satisfies these five axioms math T a,0,b T 0,a,b b quad forall a,b in R math math T 1,a,0 T a,1,0 a quad forall a in R math math forall a,b,c,d in R, a neq c math , there is a unique math x in R math such that math T x,a,b T x,c,d math math forall a,b,c in R math , there is a unique math x in R math , such that math T a,b,x c math and math forall a,b,c,d in R, a neq c math , the equations math T a,x,y b,T c,x,y d math have a unique solution math x,y in R 2 math . When math R math is finite, the third and fifth axioms are equivalent in the presence of the fourth. No other pair 0 ,1 in math R 2 math can be found such that math T math still satisfies the first two axioms. Binary operations Addition Define math a oplus b T 1,a,b math . The structure math R, oplus math turns out be a Loop algebra loop with identity element 0. Multiplication Define math a otimes b T a,b,0 math . The set math R 0 R backslash 0 math turns out be closed under this multiplication. The structure math R 0 , otimes math also turns out to be a loop with identity element 1. Linear PTR A planar ternary ring math R,T math is said to be linear if math T a,b,c a otimes b oplus c quad forall a,b,c in R math . For example, the planar ternary ring associated ... ternary ring math R,T math , one can construct a projective plane in this way math infty math is a random ... with a certain planar ternary ring. However, two nonisomorphic planar ternary rings can lead to the construction ... more details
wiktionary TCAM TCAM may refer to Telecommunications Access Method content addressable memory Ternary CAMs Ternary content addressable memory Telecommunications Conformity Assessment and Market Surveillance Committee To Kill a Mockingbird Alternative medicine Traditional, Complementary and Alternative medicine Transport Chemical Aerosol Model Disambig ... more details
Summary This is a very basic ternary diagram in svg for hypothetical components A, B, and C. I m hoping that someone perhaps me when I have time will use this base for a more thorough explanation of ternary phase diagrams. Licensing PD self date November 2007 ... more details
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