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monad    音標拼音: [m'onæd]
n. 單位,單一體,單細胞生物

單位,單一體,單細胞生物

monad
一元

monad
n 1: (chemistry) an atom having a valence of one
2: a singular metaphysical entity from which material properties
are said to derive [synonym: {monad}, {monas}]
3: (biology) a single-celled microorganism (especially a
flagellate protozoan)

Monad \Mon"ad\, n. [L. monas, -adis, a unit, Gr. ?, ?, fr.
mo`nos alone.]
1. An ultimate atom, or simple, unextended point; something
ultimate and indivisible.
[1913 Webster]

2. (Philos. of Leibnitz) The elementary and indestructible
units which were conceived of as endowed with the power to
produce all the changes they undergo, and thus determine
all physical and spiritual phenomena.
[1913 Webster]

3. (Zool.) One of the smallest flagellate Infusoria; esp.,
the species of the genus {Monas}, and allied genera.
[1913 Webster]

4. (Biol.) A simple, minute organism; a primary cell, germ,
or plastid.
[1913 Webster]

5. (Chem.) An atom or radical whose valence is one, or which
can combine with, be replaced by, or exchanged for, one
atom of hydrogen.
[1913 Webster]

{Monad deme} (Biol.), in tectology, a unit of the first order
of individuality.
[1913 Webster]

63 Moby Thesaurus words for "monad":
I, ace, air, an existence, atom, atomic particles, being, body,
brute matter, building block, chemical element, component,
constituent, creature, critter, earth, electron, element,
elementary particle, elementary unit, entelechy, entity, fire,
fundamental particle, hyle, hypostasis, individual, ion, life,
material, material world, materiality, matter, meson, molecule,
natural world, nature, no other, none else, nothing else,
nought beside, nuclear particle, object, one, one and only,
organism, person, persona, personality, physical world, plenum,
proton, quark, something, soul, stuff, substance, substratum,
the four elements, thing, unit, unit of being, water

/mo'nad/ A technique from
{category theory} which has been adopted as a way of dealing
with {state} in {functional programming languages} in such a
way that the details of the state are hidden or abstracted out
of code that merely passes it on unchanged.

A monad has three components: a means of augmenting an
existing type, a means of creating a default value of this new
type from a value of the original type, and a replacement for
the basic application operator for the old type that works
with the new type.

The alternative to passing state via a monad is to add an
extra argument and return value to many functions which have
no interest in that state. Monads can encapsulate state, side
effects, exception handling, global data, etc. in a purely
lazily functional way.

A monad can be expressed as the triple, (M, unitM, bindM)
where M is a function on types and (using {Haskell} notation):

unitM :: a -> M a
bindM :: M a -> (a -> M b) -> M b

I.e. unitM converts an ordinary value of type a in to monadic
form and bindM applies a function to a monadic value after
de-monadising it. E.g. a state transformer monad:

type S a = State -> (a, State)
unitS a = \ s0 -> (a, s0)
m `bindS` k = \ s0 -> let (a,s1) = m s0
in k a s1

Here unitS adds some initial state to an ordinary value and
bindS applies function k to a value m. (`fun` is Haskell
notation for using a function as an {infix} operator). Both m
and k take a state as input and return a new state as part of
their output. The construction

m `bindS` k

composes these two state transformers into one while also
passing the value of m to k.

Monads are a powerful tool in {functional programming}. If a
program is written using a monad to pass around a variable
(like the state in the example above) then it is easy to
change what is passed around simply by changing the monad.
Only the parts of the program which deal directly with the
quantity concerned need be altered, parts which merely pass it
on unchanged will stay the same.

In functional programming, unitM is often called initM or
returnM and bindM is called thenM. A third function, mapM is
frequently defined in terms of then and return. This applies
a given function to a list of monadic values, threading some
variable (e.g. state) through the applications:

mapM :: (a -> M b) -> [a] -> M [b]
mapM f [] = returnM []
mapM f (x:xs) = f x `thenM` ( \ x2 ->
mapM f xs `thenM` ( \ xs2 ->
returnM (x2 : xs2) ))

(2000-03-09)

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英文字典中文字典相關資料:
  • What is it that Leibniz calls a “Monad”?
    Monads capable of this are called "vernünftige Seelen" or "Geister" (intelligent souls or spirits, "animae") Now, there is a hierarchy of monads with regard to their apperceptions: From "schlummernden" (slumbering) up to the omniscient monad, the "monas monadum" (monad of monads) – God Somewhere in between are animal and spirit monads
  • What is a monad in a - Mathematics Stack Exchange
    Once you know this, then the usual definition of monad carries over straightforwardly at least in a strict 2-category In a weak 2-category (i e bicategory in the sense of Bénabou), because we need not have $(h \circ g) \circ f \stackrel{?}{=} h \circ (g \circ f)$ for 1-cells, one has to be a little bit more careful with the definition of
  • How do monads arise in mathematics? - Mathematics Stack Exchange
    It sounds like he actually introduced the notion of a comonad as opposed to a monad, and he used the term "standard construction" I'm completely guessing here since I don't have access to the article, but I suspect he was defining a comonad to organize the degeneracy and face maps of a simplicial complex or similar
  • Simple explanation of a monad - Mathematics Stack Exchange
    $\begingroup$ They are very much the same because a monad in Haskell is just an example of a category-theoretical monad in the category $\bf Hask$ (well, internal to Hask that is ) Its pretty much the same as $\bf Set$ though $\endgroup$ –
  • Leibniz monads connection to the physical universe atom
    Monad is indivisible and indestructible due to the unity (self-containment) nature As you can see, Dalton's atom is borrowed from the lemma 3 By the time Dalton was around, Leibniz's metaphysical conception of monad was far out of favor
  • What are the algebras of the double powerset monad?
    Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
  • Is the powerset monad unique? - Mathematics Stack Exchange
    Any hints if this monad already appears in the literature and if it has a name are appreciated Algebras for the monad This raises of course the question how to describe the algebras for this monad One can mimic the proof for the usual monad structure, where the algebras are sup-lattices, and arrive at those partially ordered sets where
  • Understanding the Giry monad - Mathematics Stack Exchange
    So a monad on a category $\mathbf{C}$ is basically an endofunctor on $\mathbf{C}$ together with a family of mapts $\mu_X : TTX \rightarrow TX$ that are compatible in a certain way The actual definition is a little bit more complicated; for example, we also have a monoidal unit $\eta$
  • category theory - Adjunction between a comonad and a monad . . .
    Now a monad in a 2-category $\mathcal K$ is equivalently a monad in $\mathcal K^{\mathrm{op}}$ (a monoid in a monoidal category remains a monoid when the monoidal product is reversed) and a comonad in $\mathcal K^{\mathrm{co}}$ (a monoid in a monoidal category is a comonoid in the opposite monoidal category )
  • Is monad exactly the thing that makes Kliesli arrows composable?
    Is the monad exactly the thing that makes composition work in the Kleisli category? In more detail, let $\langle T, \eta, \mu \rangle$ be a monad over a category $\mathcal C$





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