GRAPHS ASSOCIATED WITH QUANTALES

M. E. Antony and N. R. Mangalambal

  DOI:
  https://doi.org/10.37418/amsj.9.4.63

Full Text

In this paper we study the graphs associated with Quantales. Quantales are the lattice theoretic counterpart of semi rings. In literature, a wide variety of papers are available associating graphs with rings. We extend this idea to quantales. A quantale is a complete sup-lattice Q together with an associative binary operation $\ast$ satisfying the infinite distributive laws i) $a\ast (\vee_{\alpha}b_{\alpha})=\vee_{\alpha}(a\ast b_{\alpha})$ and ii) $(\vee_{\alpha}b_{\alpha})\ast a=\vee_{\alpha}(b_{\alpha}\ast a)$ for all $a\in Q$ and $ \{b_{\alpha}\} \subseteq Q$. We define a map called deductions on Quantales and develop ideals of the form $[a] ^{d}=\{x \in Q: d(x*a)=0 \} $with respect to the deduction on the quantale. We study the properties of graph associated with these ideals. We define an equivalence relation on the collection of ideals $[a] ^{d}$ and we observe that the associated graph G(Q,d) is a disconnected graph. We have also constructed the zero divisor graph $Z_{G}$ and have studied some of its properties.


Keywords:
Quantale, Deductions, Congruences, zero divisor graph.