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A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest element"?

What would mean to be "intuitionistic" for such a generalization without 1? My feeling (not to say "my intuition"), is that a counterpart of the law of excluded middle for such algebras is the equation

(x - (x - y)) = (y - (y - x)), where "-" stands for relative complement.

Did anybody read anything similar to this?

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    $\begingroup$ The topless Boolean algebras (a Boolean algebra with top element removed) arise in the characterization of the modal logic of forcing for certain forcing classes, such as in arxiv.org/abs/1207.5841. (But that use is purely classical, rather than intuistionistic.) $\endgroup$ – Joel David Hamkins Apr 11 '14 at 23:32
  • $\begingroup$ How about a Boolean ring without identity? en.wikipedia.org/wiki/Boolean_ring $\endgroup$ – François G. Dorais Apr 11 '14 at 23:57
  • $\begingroup$ Francois: You are right - a Boolean ring without identity is another "formulation" of a topless Boolean algebra. But then what would be a "pseudo-Boolean ring" without identity? @JoelDavidHamkins. Joel: That topless Boolean algebras have so close relationship with (advanced topics of) set theory is a great thing to hear, because my question is related with my interest in the universe of discourse of set theory viewed as algebra (even though such algebra would have a class as support). Unfortunately I know very little about forcing. Well, and the universe of sets is a topless Boolean algebra. $\endgroup$ – Ioachim Drugus Apr 12 '14 at 0:54
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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).

Edit. I checked "Cardinal algebras" and Wehrung's papers about POMs and I did not find this method, but I assure you that I did not invent it. Moreover, the method also works (and is used) more generally when $C$-structures are induced only on initial intervals $[0,b]$ (and not always on all intervals $[a,b]$). When $C$ is a class of lattices with additional operations which is first order definable (resp. quasi-variety, variety, finitely axiomatisable) then $C'$ also is. In this specific case: when $C$ is the class of Heyting algebas, then $C'$ is the class of distributive lattices with $0$ such that every interval $[0,a]$ is pseudo-complemented; as operations (besides lattice join and meet) one can use the constant $0$ (first order definable from the lattice, but for an equational definition it is useful to specify it in the same way as the unit is specified in the usual equational definition of groups) and the binary operation "pseudo-complement of $x$ in $x\vee y$" (a sefl-residuation operation, see Birkoff, lattice theory) i.e. the largest subelement of $x\vee y$ disjoint from $x$. When adding the constant $1$ one has then exactly Heyting algebras; equations in $C$ not using $1$ or the "total" pseudo-complement (of an element in $1$) are also valid in $C'$, and conversely each identity for $C$ can be relativized to an identity valid in $C'$ by replacing $1$ with a new variable $z$ and the absolute pseudo-complement operation by the relative pseudo-complement in $z$; each variable $x$ in the original equation is replaced by $x\wedge z$. This is general: since the initial intervals of a $C'$-structure are $C$-structures, and $C$-structures are exactly the $C'$-structures with a top element, then $C$ generates the same variety (resp. quasivariety) than $C'$ in the (topless) language of $C'$.

As essentially noted by Wouter Stekelenburg, one can do all this also starting with the dual of Heyting algebras (another class $C$ that satisfies the above conditions), and obtain another interesting variety $C'$ (even if it might not be the variety you are interested in).

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  • $\begingroup$ With this generalization method, there can exist a mapping F of tautologies in the signature of C to tautologies in the signature of C'. Is there such a method? In topless boolean algebra I found counterparts of some boolean equations this manner: if E is a boolean tautology, then I replaced in E each occurence of "-x" with "U-x", where U is the disjunction of all variables in E. Is such or other mapping used by other authors? $\endgroup$ – Ioachim Drugus Apr 12 '14 at 23:48
  • $\begingroup$ Does the generalization method you described also work when generalized structures do not contain both the bottom and top element? $\endgroup$ – Ioachim Drugus May 30 '14 at 16:09
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Because Boolean algebras are closed under duals, removing the top is equivalent to removing the bottom. There are many natural examples of duals of "bottomless Heyting algebras". Take the posets of compact subspaces of arbitrary non compact spaces for example. Bottomless Heyting algebras are models of minimal propositional logic.

Note that $p\to p = \top$. A topless Heyting algebra therefore cannot have a total pseudocomplement operator.

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    $\begingroup$ The generalized Boolean algebras that the OP is talking about always have a bottom element. The difference is that they only have a relative complement operation $x - y$ rather than an absolute complement $-x$. Thus they always have a bottom $0 = x - x$ and they are Boolean algebras precisely when they have a top $1$, in which case the complement of $x$ is the relative complement $1-x$. $\endgroup$ – François G. Dorais Apr 12 '14 at 18:14

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