The Convolution Algebra

Abstract

For a complete lattice $L$ and a relational structure $\mathfrak{X}=(X,(R_i)_I)$, we introduce the convolution algebra $L^{\mathfrak{X}}$. This algebra consists of the lattice $L^X$ equipped with an additional $n_i$-ary operation $f_i$ for each $n_i+1$-ary relation $R_i$ of $\mathfrak{X}$. For $\alpha_1,\ldots,\alpha_{n_i}\in L^X$ and $x\in X$ we set $f_i(\alpha_1,\ldots,\alpha_{n_i})(x)=\bigvee\{\alpha_1(x_1)\wedge\cdots\wedge\alpha_{n_i}(x_{n_i}):(x_1,\ldots,x_{n_i},x)\in R_i\}$. For the 2-element lattice $2$, $2^\mathfrak{X}$ is the reduct of the familiar complex algebra $\mathfrak{X}^+$ obtained by removing Boolean complementation from the signature. It is shown that this construction is bifunctorial and behaves well with respect to one-one and onto maps and with respect to products. When $L$ is the reduct of a complete Heyting algebra, the operations of $L^\mathfrak{X}$ are completely additive in each coordinate and $L^\mathfrak{X}$ is in the variety generated by $2^\mathfrak{X}$. Extensions to the construction are made to allow for completely multiplicative operations defined through meets instead of joins, as well as modifications to allow for convolutions of relational structures with partial orderings. Several examples are given.