MTL

MTL may refer to:

Science and technology

MTL (transport company)

MTL Trust Holdings was an English bus, coach and train operator based in Liverpool. MTL was originally the operating company of the Merseyside Passenger Transport Executive. To comply with the Transport Act 1985, this was divested into a new independent company, Merseyside Transport Limited (MTL). Merseytravel retained a shareholding, but this was purchased by MTL in 1993. On 17 February 2000, MTL was purchased by Arriva.

Railway operations

MTL was awarded two railway franchises in 1997, during the Privatisation of British Rail. Both franchises were taken over by Arriva in 2000.

Northern Spirit was a large franchise operating services across the whole of Yorkshire, including the Settle & Carlisle line and much of the North East. The franchise also included TransPennine Express trains from Liverpool and Blackpool to Scarborough, Hull and Newcastle. The franchise was rebranded as Arriva Trains Northern in April 2001.

Merseyrail Electrics was responsible for running the Merseyrail network in Merseyside. All services were subsidised by Merseytravel. The franchise was rebranded as Arriva Trains Merseyside in April 2001.

Monoidal t-norm logic

Monoidal t-norm based logic (or shortly MTL), the logic of left-continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of commutative bounded integral residuated lattices (known as Höhle's monoidal logic, Ono's FL_ew, or intuitionistic logic without contraction) by the axiom of prelinearity.

Motivation

T-norms are binary functions on the real unit interval [0, 1] which are often used to represent a conjunction connective in fuzzy logic. Every left-continuous t-norm has a unique residuum, that is, a function such that for all x, y, and z,

The residuum of a left-continuous t-norm can explicitly be defined as

This ensures that the residuum is the largest function such that for all x and y,

The latter can be interpreted as a fuzzy version of the modus ponens rule of inference. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold.