A Heyting lattice $L$ is a Brouwer-intuitionistic logic lattice with a bottom, or lowest element $0$. In the more technical classification it is a commutative lattice which is both ‘pseudocomplemented and also relatively pseudocomplemented’. The concept of relative pseudocomplementation coincides with the material implication operator, $\Rightarrow$, in symbolic propositional logic based on chryssippian or Boolean logic.

A Heyting algebra is a p-algebra (as defined next in Definition 1.3 ) with the relative pseudocomplentation operation $\to$ (which replaces the propositional implication $\Rightarrow$).
Given an element $a$ in a bounded lattice $L$, a complement of $a$ is defined to be an element $b\in L$, if such an element exists, such that

To surmount the non-uniqueness of the complement, an alternative to the latter was defined– the pseudocomplement of an element.

An element $b$ in a lattice $L$ with $0$ is a pseudocomplement of $a\in L$ if

1. 1.

   $b\wedge a=0$

2. 2.

   for any $c$ such that $c\wedge a=0$ then $c\leq b$.

In other words, $b$ is the maximal element in the set $\{c\in L\mid c\wedge a=0\}$.

A convenient modification of the pseudocomplemented (pc) lattice concept is a p-algebra (or pseudocomplemented algebra) which is a pc-lattice where ${}^{*}$ is regarded as an algebraic operator. Thus, a morphism of pc–lattices is a proper lattice homomorphism, whereas a morphism between two p-algebras is a lattice homomorphism $f$ that also preserves the pc-algebraic operation ${}^{*}$, i.e., $f(a^{*})=f(a)^{*}$. One can therefore define a category of p-algebras by specifying the morphism between any pair of p-algebras (considered as objects of this algebraic logic category) as the $\{0,1\}$-lattice homomorphism, with the following condition $f(1)=f(0^{*})=f(0)^{*}=0^{*}=1$ being also satisfied.