The complex numbers and other field extensions The set of complex numbers , i.e., numbers that can be written in the form for real numbers and where is the imaginary unit, form a vector space over the reals with the usual addition and multiplication and for real numbers , , , and .

The underlying set of the group is the above set of symmetry functions, and the group operation is function composition.

For example, consider the natural numbers with addition as the operation.

The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends.

Projection () A projection is a unary operation written as formula_1 where formula_2 is a set of attribute names.

For example, operation preserving functions exist between the set of real numbers with addition and the set of positive real numbers with multiplication.

For every non empty set S there is a binary operation defined on S that gives it a group structure.

In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element.

Member States shall ensure that a payment transaction is considered to be authorised only if the payer has consented to the respective payment order addressed to the payment service provider .

Consent shall consist in an explicit authorisation for the payment service provider to effect a payment transaction or a series of transactions .

For example, a symmetry group encodes symmetry features of a geometrical object the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other.

Purchase date the date on which the sale of purchased assets by the seller to the buyer becomes effective .

Announcement of longer term refinancing operation Allotment of longer term refinancing operations Settlement of longer term refinancing operations

Even though the set may be the same, the same function might be a group homomorphism, (a single binary operation, an inverse operation, being a unary operation, and identity, being a nullary operation) but not a ring isomorphism (two binary operations, the additive inverse and the identity elements), because it may fail to preserve the additional monoid structure required by the definition of a ring.

Interest rate instruments are revalued on an item by item basis .

ECB Annual Report 2006 Interest rate instruments are revalued on an item by item basis .

Interest rate instruments are revalued on an itemby item basis .

Interest rate instruments are revalued on an item by item basis .

In addition , the EU Council may meet in the composition of the Heads of State or Government .

Outright transaction a transaction where the central bank buys or sells assets outright in the market ( spot or forward ) .

Each can be defined in essentially two different ways either by constructing an explicit well ordered set that represents the operation or by using transfinite recursion.

Quick tender the tender procedure used by the Eurosystem for fine tuning operations when it is deemed desirable to have a rapid impact on the liquidity situation in the market .

Main refinancing operation a regular open market operation executed by the Eurosystem in the form of a reverse transaction .

Main refinancing operations are conducted through weekly standard tenders and have a maturity of two weeks . Maintenance period the period over which credit institutions compliance with reserve requirements is calculated .

Pooling system a system for central banks collateral management where the counterparties provide a pool of assets to serve as collateral in their transactions with the central bank .

As for the instruments used , reverse transactions are the main open market instrument of the Eurosystem and can be employed in all four categories of operations .

When , as sometimes happens , a single operation in securities ( perhaps connected with a takeover or merger ) is carried out in stages over more than one month , the practice is to record the whole operation in the month in which it is completed .

The single instruction multiple data (SIMD) classification is analogous to doing the same operation repeatedly over a large data set.

Euro area the area encompassing those Member States in which the euro has been adopted as the single currency in accordance with the Treaty and in which a single monetary policy is conducted under the responsibility of the relevant decision making bodies of the ECB . The euro area comprises Belgium , Germany , Spain , France , Ireland , Italy , Luxembourg , the Netherlands , Austria , Portugal and Finland .

Non financial corporations one of the institutional sectors in the European System of Accounts 1995 ( ESA 95 ) .

the type of operation

Monetary policy instrument

the type of operation

Longer term refinancing operation a regular open market operation executed by the Eurosystem in the form of a reverse transaction .

In addition, the Cartesian product is defined differently from the one in set theory in the sense that tuples are considered to be shallow for the purposes of the operation.

By contrast with an earmarking system , the underlying assets are not earmarked for individual transactions in a pooling system . Post fixed coupon a coupon on floating rate instruments which is determined on the basis of the values taken by the reference index at a certain date ( certain dates ) during the coupon accrual period .

In some Member States , branches of the national central bank may be closed on NCB business days owing to local or regional bank holidays .

Post fixed coupon a coupon on floating rate instruments which is determined on the basis of the values taken by the reference index at a certain date ( or dates ) during the coupon accrual period .

Pooling system see collateral pooling system .

Minimum bid rate the lower limit to the interest rates at which counterparties may submit bids in variable rate tenders .

Post fixed coupon a coupon on floating rate instruments which is determined on the basis of the values taken by the reference index on a certain date ( or dates ) during the coupon accrual period .

National central bank ( NCB ) as used in this document , a central bank of an EU Member State which has adopted the single currency in accordance with the Treaty .

Outright transaction a transaction whereby assets are bought or sold up to their maturity ( spot or forward ) .

Outright transaction a transaction whereby assets are bought or sold up to their maturity ( spot or forward ) .

ECBc The implementation of monetary policy in the euro area February 2005 ANNEX 2 Outright transaction a transaction whereby assets are bought or sold up to their maturity ( spot or forward ) .