رسته ها و ساختارهای کلی جبری با کاربردها
سال:1396 | دوره:7 | شماره:1 |تعداد مقالات:12

Professor Bernhard Banaschewski was born in Munich on 22nd March 1926. He studied at the University of Hamburg until 1953, receiving his doctorate in Mathematics under Ernst Witt. The political situation in Hitler era made him to move to McMaster University in Hamilton, Ontario, Canada, in 1955. There he made rapid progress and was awarded the prestigious McKay Professorship of Mathematics in 1964.

This interview, a co-operative effort of Bernhard Banaschewski and Christopher Gilmour, took place over a few days in December, 2016. It was finalised over coffee and a shared slice of excellent cheesecake atThe Botanical Tea Garden, a small, home situated, tea garden in Little Mowbray, Cape Town.

The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense.This operation models a logical ‘tangle modality’ connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation.

In this paper, for eachlattice-valued map A®L with some properties, a ring representation A®RL is constructed. This representation is denoted byc which is an f-ring homomorphism and a Q-linear map, where its indexc, mentions to a lattice-valued map. We use the notation a dpqa= (a − p)+^ (q − a)+, where p, q Î Q and a Î A, that is nominated asinterval projection. To get a well-defined f-ring homomorphism tc, we need such concepts asbounded, continuous, and Q-compatible for c, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-valued map c f: A®L for each f-ring homomorphism f: A®RL. It is proved that crc=cr and tcf=f, which they make a kind of correspondence relation between ring representations A®RL and the lattice-valued maps A®L, where the mapping cr: A®L is called a realizationof c. It is shown that tcr=tc and crr=cr.