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In this paper, we study and characterize some properties of a given binary operation on a lattice. More specifically, we show necessary and sufficient conditions under which a binary operation on a lattice coincides with its meet (resp. its join) operation. Importantly, we construct two new posets based on a given binary operation on a lattice and investigate some cases that these two posets have a lattice structure. Moreover, we provide some representations of a given lattice based on these new...
All ordinal numbers with the following property are found: there exists a loop such that its subloops form a chain of ordinal type .
This paper aims to propose a complete relational semantics for the so-called logic of bounded lattices, and prove a completeness theorem with regard to a class of two-sorted frames that is dually equivalent (categorically) to the variety of bounded lattices.
Let R be an associative ring with 1 and R-tors the somplete Brouwerian lattice of all hereditary torsion theories on the category of left R-modules. A well known result asserts that R is a left semiartinian ring iff R-tors is a complete atomic Boolean lattice. In this note we prove that if L is a complete atomic Boolean lattice then there exists a left semiartinian ring R such that L is lattice-isomorphic to R-tors.
We define and study classification systems in an arbitrary CJ-generated complete lattice L. Introducing a partial order among the classification systems of L, we obtain a complete lattice denoted by Cls(L). By using the elements of the classification systems, another lattice is also constructed: the box lattice B(L) of L. We show that B(L) is an atomistic complete lattice, moreover Cls(L)=Cls(B(L)). If B(L) is a pseudocomplemented lattice, then every classification system of L is independent and...
An M-Set is a unary algebra whose set of operations is a monoid of transformations of ; is a G-Set if is a group. A lattice is said to be represented by an M-Set if the congruence lattice of is isomorphic to . Given an algebraic lattice , an invariant is introduced here. provides substantial information about properties common to all representations of by intransitive G-Sets. is a sublattice of (possibly isomorphic to the trivial lattice), a -product lattice. A -product...
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