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Quasimodules over bounded lattices

Ivan Chajda, Helmut Länger

Abstract

We define a quasimodule Q over a bounded lattice L in an analogous way as a module over a semiring is defined. The essential difference is that L need not be distributive. Also for quasimodules there can be introduced the concepts of inner product, orthogonal elements, orthogonal subsets, bases and closed subquasimodules. We show that the set of all closed subquasimodules forms a complete lattice having orthogonality as an antitone involution. Using the Galois connection induced by this orthogonality, we describe important properties of closed subquasimodules. We call a subquasimodule P of a quasimodule Q splitting if the sum of P and its orthogonal companion is the whole set Q and the intersection of P and its orthogonal companion is trivial. We show that every splitting subquasimodule is closed and that its orthogonal companion is splitting, too. Our results are illuminated by several examples.

Quasimodules over bounded lattices

Abstract

We define a quasimodule Q over a bounded lattice L in an analogous way as a module over a semiring is defined. The essential difference is that L need not be distributive. Also for quasimodules there can be introduced the concepts of inner product, orthogonal elements, orthogonal subsets, bases and closed subquasimodules. We show that the set of all closed subquasimodules forms a complete lattice having orthogonality as an antitone involution. Using the Galois connection induced by this orthogonality, we describe important properties of closed subquasimodules. We call a subquasimodule P of a quasimodule Q splitting if the sum of P and its orthogonal companion is the whole set Q and the intersection of P and its orthogonal companion is trivial. We show that every splitting subquasimodule is closed and that its orthogonal companion is splitting, too. Our results are illuminated by several examples.

Paper Structure

This paper contains 14 theorems, 21 equations.

Key Result

Lemma 4

Let $\mathbf L=(L,\vee,\wedge,0,1)$ be a bounded lattice, $I$ a non-empty index set, for every $i\in I$ let $L_i$ be a non-empty ideal of $\mathbf L$, put $Q:=\prod\limits_{i\in I}L_i$ and $\vec{0}:=(0,\ldots,0)$ and define $+\colon Q\times Q\to Q$ and $\cdot\colon L\times Q\to Q$ by for all $(a_1;i\in I),(b_i;i\in I)\in Q$ and all $c\in L$. Then $\prod\limits_{i\in I}\mathbf L_i:=(Q,+,\cdot,\vec

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Example 3
  • Lemma 4
  • Definition 5
  • Definition 6
  • Theorem 7
  • proof
  • Definition 8
  • Remark 9
  • ...and 31 more