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Lattice Theory
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
any semilattice with binary operation ∘ becomes a partially ordered set in which x°y=lub{x,y}. It should now be obvious that if S is a partially ordered set for which the operation x∧y is defined for each pair x,y∈S, then S is also a semilattice. Whenever the operation x∨y or x∧y are known or assumed, then the notation (S,∨) or (S,∧) will be used to specify these semilattice in order to avoid any ambiguities. Some authors refer to the semilattices (S,∨) and (S,∧) as the join and meet semilattices, respectively. In this text, however, the semilattice (S,∨) will be referred to as a max-semilattice and (S,∧) as a min-semilattice.
On pattern setups and pattern multistructures
Published in International Journal of General Systems, 2020
Aimene Belfodil, Sergei O. Kuznetsov, Mehdi Kaytoue
If infimum and supremum of S exist, then Moreover, since for a poset we have , then the empty set has a meet (resp., join) if and only if the poset is upper-bounded (resp., lower bounded) and we have and . A poset is said to be A (complete) meet-semilattice if any pair (resp., any subset) of elements of P has a meet.A (complete) join-semilattice if any pair (resp., any subset) of elements of P has a join.A lattice if it is both meet-semilattice and join-semilattice with respect to the same order relation.A complete lattice if all its subsets including ∅ have meets and joins.
Logics of variable inclusion and the lattice of consequence relations
Published in Journal of Applied Non-Classical Logics, 2020
For standard information on Płonka sums, we refer the reader to Płonka (1967a, 1967b), Płonka and Romanowska (1992), and Romanowska and Smith (2002). A semilattice is an algebra , where ∨ is a binary commutative, associative and idempotent operation. Given a semilattice and , we set It is easy to see that < is a partial order on A.