The general method to consider "unitless" order structures is the following:

suppose that $C$ is a class of structures such that on each structure $X$ in $C$ a natural order with bounds $0_X,1_X$ is defined, and each interval $[a,b]\subseteq X$ for that order again has a natural $C$-structure, and finally this "induced structure" construction is transitive (the natural order in $[a,b]$ as $C$-structure is the one induced by $X$ as poset, and on any subinterval of $[a,b]$ the induced $C$-structure and order as interval in $X$ or in $[a,b]$ coincide).

Boolean algebras are an example of such class $C$, but Heyting algebras also are (other examples: complemented modular lattices, orthomodular lattices and posets, orthoalgebras, D-posets, finite length lattices, and many more classes of lattices). Then the natural generalization of $C$ to a class $C'$ where top element need not exist is: the class of sup directed posets with bottom element $0$ such that on each interval $[0,a]$ a $C$-structure is given and when $a\leq b$ this structure is the one induced by the $C$-structure on $[0,b]$. 

When on each $C$-structure the order is a join semilattice (resp. lattice), then the same happens for $C'$-structures (note the sup-directed request above; a further generalization without such request is possible but then the "coherence" axiom must be taken in a stronger form)

Every example that I know in lattice theory for "generalized structures without 1" follows this pattern. Surprisingly, I cannot find this in Gratzer book, but I remember that Tarski wrote something about this general method (perhaps in "Cardinal algebras", but I have not checked this in the last 20 years). Perhaps also F. Wehrung papers around 1993 contain something around the same lines (again, these are recollections of about 20 years ago; since then I have always used this general method without ever checking the original sources and now I have forgotten the precise references).