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consider a classical constrained theory @xcite , with configurations @xmath0 , conjugate momenta @xmath1 , classical brackets @xmath2|}$ ] and constraints @xmath3 . _ observables _ @xcite are then objects forming zero brackets with the constraints , _ o _ 0 . [ basic - obs ] there are multiple such notions due to ` zero ' , ` brackets ' and ` the constraints ' taking a variety of different precise meanings @xcite . some such notions of observables can be more physically useful than just any functions ( or functionals ) of @xmath0 and @xmath1 . this is because of their greater physical content ; moreover , at least some versions of such notions of observables would be expected to contain physical information only . at the very least this property is required in phrasing final answers to physical questions about a theory . the zero can for instance be the usual notion of zero  alias _ strongly zero _ in this context  or dirac s notion @xcite of _ weakly zero _ , denoted @xmath4 , meaning zero up to terms linear in constraints . the bracket can for instance , at the classical level , be the usual poisson bracket ; we restrict attention to this case in the rest of this article . the constraints can include all the constraints , or a subset thereof . three particular cases of this which are well - known enter the current article . \1 ) _ classical dirac observables _ @xcite are quantities @xmath5 that ( usually weakly poisson ) brackets - commute with _ all _ of a given theory s first - class constraints , _ d _ 0 . [ c - d ] various interpretations proposed for these entities gave further colourful names for these , such as ` evolving constants of the motion ' @xcite and ` perennials ' @xcite . the names ` true ' and ` complete ' observables have also been used in similar contexts @xcite . conceptually , these are ` _ maximally constrained _ ' observables ; the current article will also give mathematical cause to call these ` _ zeroth observables _ ' . \2 ) ` _ unconstrained observables _ ' @xmath6 , on the other hand have no constraint - related restrictions . ` _ unital observables _ ' is also a mathematical name for these for reasons which will become clear as this article develops . ` partial observables ' @xcite are a well - known particular notion of unconstrained observables ( though these play no further part in the current article ) . \3 ) introduced @xcite a further notion of observables , which subsequently indeed came to be termed _ observables_. these are quantities @xmath7 which ( again usually weakly poisson ) brackets - commute with all of a given theory s first - class linear constraints , _ k _ 0 . [ flin - k ] using observables reflects treating first - class quadratic constraints , @xmath8 , distinctly from first - class linear ones @xmath9 @xcite . the problem of time literature @xcite is replete with reasons why this may be desirable ; moreover also observables are more straightforward to find than dirac ones . indeed , the _ problem of observables _  that it is hard to construct a sufficiently large set of these to describe all physical quantities , especially for gravitational theory is one of the many facets of the problem of time . observables are more straightforward to construct than dirac ones , and classical ones are more straightforward to construct than quantum ones . the problem of observables remains little understood . strategies proposed in this regard include the following . strategy 1 ) insist upon constructing dirac observables , @xmath5 . strategy 2 ) make use of unconstrained observables , @xmath6 @xcite .    in strategy 2 ) case one is disregarding all the information content of the constraints . however , on some occasions strategy 2 ) is used as a stepping stone to constructing dirac - type observables . strategy 3 ) consider observables , @xmath7 , to suffice . some examples are also useful at this stage . in unconstrained theories , all three notions coincide . in theories whose sole constraint is first - class quadratic , unconstrained and observables are coincident , trivial and distinct from dirac observables , which are nontrivial . this applies to minisuperspace models and to some of the simplest relational mechanics @xcite models . in electromagnetism and yang mills theory , on the other hand , all the constraints are first - class linear , so dirac and observables coincide and are distinct from unconstrained observables . finally , in more advanced relational mechanics theories @xcite and in gr , there are first - class linear constraints and a first - class quadratic constraint . here , none of the above three notions of observables coincide . in particular , for gr the first - class linear constraints are the vector - valued gr momentum constraint @xmath10 , whereas the first - class quadratic constraint is the scalar - valued gr hamiltonian constraint @xmath11 . the observables then obey _ i k _ 0 , [ k_gr ] where @xmath12 is specifically the poisson bracket . for relational mechanics , @xmath13 which has the mathematical form of a quadratic energy constraint takes over the role of @xmath11 . here many examples observables are known explicitly @xcite : they are shapes , or shapes and scales , for various different notions of shape @xcite , and conjugate quantities . for electromagnetism , the observables forms are also well - understood @xcite , however for gr they remain only formally understood @xcite . the next consideration of note is dirac s conjecture , by which first - class linear constraints would always coincide with gauge constraints . unfortunately , this is false @xcite . due to this , a fourth notion of observable is unearthed : the @xmath14-observables @xmath15 which commute with all of a theory s gauge constraints : g _ g _ 0 ; @xmath14 is that gauge theory s underlying gauge group . of course , for some theories @xmath9 and @xmath16 coincide , in which case @xmath7 and @xmath15 coincide as well ; e.g. this is the case in relational mechanics , electromagnetism and yang  mills theory .    strategy 4 ) is then to consider @xmath14-observables observables to suffice . the next development follows from perceiving that a notion of observables being meaningful is contingent on the subset of constraints it commutes with a fortiori closing as a subalgebraic structure @xcite ( lemma 1 of appendix [ observables ] ) . in particular , in some cases the @xmath9 do not close , and thus indeed the notion of observables @xmath7 does not exist for that theory . in problem of time considerations , this means that many approaches to the problem of observables can only be addressed once the constraint closure problem has been resolved @xcite . i.e. one needs the constraints algebraic structure prior to consideration of associated observables algebraic structures . it is also required that the notion of bracket involved in defining the @xmath17 matches that under which the @xmath3 close . additionally , by appendix [ observables ] s lemma 2 , the @xmath17 themselves form a closed algebraic structure . [ it is , moreover , interesting that observables algebraic structures exhibit some conceptual similarities with theory of casimirs : appendix [ associated ] , which proved to be a central ingredient in understanding ( much simpler ) algebraic structures : lie groups and their reps . ]    along the above lines , focus shifts from observables to the observables associated with whichever closed subalgebraic structures each theory s constraint algebraic structure happens to possess . i term these a - observables , where the ` a ' stands for ` algebraic substructure ' . the subalgebraic structures of a gives algebraic structure then turn out to form a bounded lattice ( appendix [ lattices ] ) under the inclusion operation . the association by ` forms zero brackets with ' then produces in a further lattice , now of notions of observables . this pair of bounded lattice structures form the current article s main underlying result . fig [ latt-1 ] gives preliminary versions of this involving @xmath6 , @xmath5 , @xmath7 and @xmath15 . the current article illustrates this with nontrivial examples of notions of a - observable arising in relational mechanics , slightly inhomogeneous cosmology and supergravity , in secs 2 , 3 and 4 respectively . we conclude in sec 4 with a more general account of the lattice of constraint subalgebraic structures and of the associated lattice of notions of a - observables . discussion of the article s range of algebraic concepts is difered to a multi - part appendix . the constraint algebraic structures for these are outlined appendix [ cas ] . some of the constraint subalgebras are then given in fig [ latt-2 ] , alongside the associated notions of a - observables . in this model @xcite @xmath18 , and one splits modewise ( each mode corresponds to collection of @xmath19-harmonics labels ) and into scalar , vector and tensor ( svt ) parts . this splits @xmath11 into @xmath20 , @xmath21 and @xmath22 pieces , and @xmath10 into @xmath23 and @xmath24 pieces . modewise and svt - wise splits give decoupled equations at the unreduced level , to lowest nontrivial perturbative order . it then turns out that @xcite taking out @xmath23 and @xmath24 fails to take out all of the @xmath25 information left in the model . the amount by which it fails to do so depends on whether one is considering the single minimally coupled scalar field matter version @xcite or the vacuum version @xcite . in the vacuum case , it is out by 1 configuration space degree of freedom , whereas it is out by 2 in the scalar field case . the vacuum case also has a more decoupled partly reduced system , by which it is clear @xcite that removing @xmath21 suffices to take one down to @xmath25 invariance , i.e. to this model s counterpart of @xmath26 . it is this vacuum case that the current article considers in detail ; the scalar field counterpart remains only partly understood . in this vacuum case , then , the constraint algebraic structure is as in fig [ latt-3].a ) and the corresponding notions of observables are as in fig [ latt-3].b ) . firstly , note that observables and @xmath14-observables  for @xmath27  are distinct in this example . i also term the latter superspace observables . secondly , note that there are additional notions of a - observables , for which i suggest the names s - observables and v - observables , meaning commutation with just @xmath23 and just @xmath24 respectively . taking out just @xmath24 has a scalar field case parallel which underlies wada s even more partial reduction in @xcite ; he then proceeded to reduce out @xmath23 as well @xcite . introducing a local lorentz frame formulation to accommodate the model s fermionic species causes a local lorentz frame constraint @xmath28 ( and its conjugate ) to appear . the canonical formulation of supergravity then has the expected @xmath11 , @xmath10 and @xmath28 constraints alongside a specifically supersymmetric constraint @xmath29 . all we need to know for this article about @xmath28 and @xmath29 are that each of these constraints is linear in the momenta , and the schematic form of the subsequent constraint brackets , as per appendix [ cas ] . the particular feature this article concentrates upon is that in supergravity a subset of the linear constraints  the supersymmetry constraints  have the supergravity counterpart of the quadratic @xmath30 as _ their _ integrability ; this follows from eq ( [ s - s->h ] ) . this means that observables are not well - defined for supergravity , since the quadratic @xmath11 now arises as an integrability of the linear @xmath29 . eq ( [ s - s->h ] ) forms the second integrability of the ` two - way ' pair , to the first integrability being of form ( [ dirac - algebroid ] ) . some constraint subalgebraic structures of note are then provided in fig [ latt-3 ] . in particular , the non - supersymmetric linear constraint combination of @xmath28 and @xmath10 closes , as does the gr - like combination of @xmath11 and @xmath10 , and the non - supersymmetric combination of @xmath28 , @xmath10 and @xmath11 . i term the corresponding notions of a - observables non - supersymmetric , gr - likes and non - supersymmetric dirac @xcite . there is a notion of observables associated with each constraint subalgebraic structure that a given theory s constraint algebraic structure happens to possess . i term these , collectively , a - observables ( fig [ latt-5 ] ) . both the constraint subalgebraic structures of a given theory s constraint algebraic structure , and the theory s notion of a - observables form bounded lattice structures . the dirac observables @xmath5 correspond to the zero of the latter lattice , arising from considering the whole constraint algebraic structure , which is the unit of the former lattice . on the other hand , the unconstrained observables @xmath6 correspond to the unit of the latter lattice , arising from considering the trivial subalgebraic structure ( i d ) , which is the zero of the former lattice . once this context has been framed , it is clear that the notion observables  commuting with first - class linear constraints  is only one some occasions well - defined . this becomes seen , rather , as an early example of nontrivial a - observable ( i.e. neither unconstrained nor dirac ) , which is realized in some theories  such as gr  but not others  such as supergravity . thus approaches to quantum gravity which depend on the quadratic to linear distinction between constraints , and the subsequent notion of observables , can be fragile to change of theory . this is to be contrasted with how @xmath5 and @xmath6 are defined for _ all _ theories . moreover , other interesting closed subalgebraic structures arise , each with an associated notion of a - observable . in the current article , we have discussed a number of such , and coined names for them . \a ) frame observables , non - supersymmetric observables , gr - like observables and non - supersymmetric dirac observables in the case of supergravity . \b ) superspace observables as distinct from observables in slightly inhomogeneous cosmology , alongside scalar - only and vector - only observables . \c ) a wide range of geometrically meaningful a - observables in the context of relational theories of mechanics .    the introduction s list of strategies toward dealing with the problem of observables can then be updated to include the following .    strategy 5 ) conceive in terms of the lattice of notions of a - observables @xmath31 , such that a _ c _ ` = ' 0 , for @xmath32 whichever ( closed ) subalgebraic structure of the constraint algebraic structure . this strategy covers    \1 ) making whichever such choice . \2 ) the possibility of having further selection principles among the various candidate theories whose constraint algebraic structures admit a variety of nontrivial proper subalgebraic structures . \3 ) using one or more notions of a - observables as intermediate stages in computing more restrictive types of observables . note moreover that the dirac observables strategy is both _ always available _ and _ requires no choice or further selection principle_. this is a direct consequence of taking the whole constraint algebraic structure , as follows . the same two maxims also apply at the opposite extreme of the lattice of subalgebraic structures : the unconstrained observables . i end by pointing out two future research directions . \1 ) increase the scope of the examples used to include more supersymmetric relational mechanics as well as superconformal supergravity . \2 ) quantum mechanical counterparts and quantization process from classical to quantum versions remain to be presented @xcite . * acknowledgements * i thank those close to me . i thank the fqxi and damtp for travel money and hosting in 2014 - 2015 ( slightly inhomogeneous cosmology ) and 2015 - 2016 ( supergravity ) . i also thank jeremy butterfield , chris isham , malcolm maccallum , don page , enrique alvarez , john barrow , marc lachieze - rey and reza tavakol for discussions or helping me with my career . @xmath14 @xcite are simultaneously groups and differentiable manifolds ; additionally their composition and inverse operations are differentiable . working with the corresponding infinitesimal ` tangent space ' around @xmath14 s identity element  the lie algebra @xmath33 is more straightforward due to vector spaces tractability , while very little information is lost in doing so . for instance , the representations of @xmath33 determine those of @xmath14 .    more formally , a _ lie algebra _ is a vector space equipped with a product ( bilinear map ) @xmath34| } : \frg \times \frg \longrightarrow \mbox{\frg}$ ] that is antisymmetric and obeys the leibniz ( product ) rule and the _ jacobi identity _ g_1 g_2 g_3 + = 0 [ jacobi - id ] @xmath35 . this an example of _ algebraic structure _ : equipping a set with one or more further product operations . particular subcases of lie brackets include the familiar poisson brackets and quantum commutators .    moreover , a lie algebra s generators ( lie group generating infinitesimal elements ) @xmath36 obey _ _ = c^ _ _ , [ str - const ] where @xmath37 are the _ structure constants _ of that lie algebra . it readily follows that the structure constants with all indices lowered are totally antisymmetric , and also obey _ [ c^ _ ] = 0 . [ firstjac ] next suppose that a hypothesis is made about some subset @xmath38 of the generators , @xmath39 , being significant . these are linear and quadratic constraints in the context of constraints in gravitational theory . denote the rest of the generators  which form @xmath40  by @xmath41 . on now needs to check to what extent the algebraic structure in question actually complies with this assignation of significance . such checks place limitations on how generalizable some intuitions and concepts which hold for simple examples of algebraic structures are . in general , the split algebraic structure is of the form k _ k_^ = c^^_^k_^ + c^_^ h _ , [ lie - split-1 ] k _ h _ = c^^_k_^ + c^^_h_^ , [ lie - split-2 ] h _ h_^ = c^_^ k _ + c^^_^h_^ . [ lie - split-3 ] denote the second to fifth right - hand - side terms by 1 ) to 4 ) . 1 ) and 4 ) being zero are clearly subalgebra closure conditions . 2 ) and 3 ) are ` interactions between ' @xmath40 and @xmath38 . the following cases of this are then realized in this article . \i ) _ direct product _ if 1 ) to 4 ) are zero , then @xmath42 : the _ direct product group _ : the cartesian product of the sets of @xmath38 and @xmath40 , with group operation @xmath43 . in this case , one can trivially make use of the representation theory of @xmath38 and @xmath40 to build that of @xmath44 . \ii ) _ semi - direct product _ if 2 ) alone is nonzero , then @xmath45 . now , @xmath46 is given by @xmath47 for @xmath48 , @xmath49 a subgroup of @xmath14 and @xmath50 a group homomorphism . in this case , _ mackey theory _  an advanced type of induced representations method @xcite can be used to construct the representation theory of semidirect product groups . \iii ) ` _ thomas integrability _ ' . if 1 ) is nonzero , then @xmath38 is not a subalgebra : attempting to close it leads to some @xmath39 are discovered to be integrabilities . i denote this by @xmath51 . a simple example of this occurs in splitting the lorentz group s generators up into rotations and boosts : the group - theoretic underpinning @xcite of thomas precession schematically , decompose the familiar lorentz algebra bracket into j j ~j , j k ~k , k k ~k + j , the key bracket being the last one by which the boosts are not a subalgebra . thomas precession then refers to the rotation arising thus from a combination of boosts . \iv ) ` _ two - way integrability _ ' . if 1 ) and 4 ) are nonzero , neither @xmath38 nor @xmath40 are subalgebras , due to their imposing integrabilities on each other . i denote this by @xmath52 . in this case , any wishes for @xmath38 to play a significant role by itself are almost certainly dashed by the actual mathematics of the algebraic structure in question . one enters the set of constraints in one s possession into the brackets in use to form a constraint algebraic structure . this may enlarge one s set of constraints , or cause of one to adopt a distinct bracket . if inconsistency is evaded , the eventual output is an algebraic structure for all of a theory s constraints , symbolically    _ _ ^ _ = c^^_^_^ . in some cases , the @xmath53 are a lie algebra s structure constants , whereas in other cases  in particular for gr , they are a lie algebroid s structure functions . see appendix [ algebroids ] for a bit more about algebroids . for now , we note that the preceding sec s split classification extends to algebroids as well . all of the above carries over to supersymmetric versions as well . examples of ii ) relational mechanics provide a number of examples of semi - direct product constraint algebras . there are versions with constraints in correspondence with each of the following : the euclidean group @xmath54 , the similarity group @xmath55 , the equiareal , ( or equivoluminal or higher-@xmath56 counterpart ) group @xmath57 , and the affine group @xmath58 . @xmath59 correspond to the zero total momentum constraint @xmath60 , @xmath61 to the zero total angular momentum constraint @xmath62 ( or arbitrary-@xmath56 generalization thereof ) , and @xmath63 to the zero total dilational momentum constraint @xmath64 . [ we are here using @xmath65 as an index running over the particle labels . ] @xmath66 corresponds to @xmath67 alongside zero total shear momentum constraints and zero total ` procrustes stretch ' ( @xmath56-volume preserving stretch ) momentum constraints @xcite , whereas @xmath68 corresponds to all of these alongside @xmath69 .    in each case , a suitably compatible form of quadratic energy constraint @xmath13 can be adjoined , but the preceding structures persist as subalgebras of first - class linear constraints . i.e. f ~f , e ~0 , e ~0 . the first bracket indicates a closed subalgebra , the second that @xmath13 is a good @xmath70 object ( a scalar ) , and the third indicates that @xmath13 forms its own subalgebra rather than implying any of the @xmath70 as integrabilities . thus adjunction of @xmath13 is a first example of i )    further examples of i ) include a further relational mechanics a theory @xcite with @xmath71 invariance which models the absense of absolute axes of rotation and absolute scale whilst retaining an absolute origin  provides an example of direct product constraint algebra of @xmath67 and @xmath69 . of course , the standard model also involves a direct product lie group @xmath72 , so @xmath73 is well - exemplified within conventional gauge theory .    examples of iii ) at the level of constraint algebraic structures include ( as outlined in @xcite ) the relational mechanics that are based on each of the conformal group @xmath74 and on all of the supersymmetric counterparts of the groups in example ii ) . the key bracket here is an integrability of the form    p ~m + d , [ k - p ] by which the conformal algebra is @xmath75 thomas . i.e. a translation and a special conformal transformation compose to give both a rotation ( ` conformal precession ' ) _ and _ an overall expansion . then another salient example of iii ) is the dirac algebroid formed by the constraints of gr @xcite , schematically _ i m_i ~m_i , _ i h ~h , h ~m . [ dirac - algebroid ] it is the last of these brackets which contains structure functions , with teitelboim pointing to the interpretational complications of this @xcite . moreover , the physics of the problem in hand requires this enlargement and complication of algebraic structure , to model the myriad of possible foliations of spacetime by spatial hypersurfaces . also note that the first of these brackets shows that the momentum constraint @xmath10 closes by itself al a subalgebraic structure ; these are first - class linear constraints corresponding to @xmath76 . it is , moreover a bona fide ( if infinite-@xmath56 ) lie algebra , of the spatial diffeomorphisms @xmath76 on the corresponding topological manifold @xmath77 . finally , the second of these brackets means that @xmath11 is a good @xmath76 object : a scalar density . example of iv ) the constraint superalgebroid of supergravity . at the level of algebraic structures , each of supersymmetry and split space - time gr can be separately envisaged as thomas - type effects . upon considering both at once , these integrabilities furthermore go in opposite directions , so the more complicated ` two - way ' integrability case arises . the schematic form of the key new relation for supergravity is s ~h ( + m + j ) . [ s - s->h ] the integrability pointing in the opposite direction remains of the schematic gr form ( [ dirac - algebroid ] ) . more details of this superalgebroid can be found at e.g. @xcite . superalgebroids are treated in outline in e.g. @xcite ; the above outline readily continues to carry over upon replacing ` lie algebra ' by ` lie superalgebra ' ; supergravity s constraint superalgebroid is probably theoretical physics most salient example of superalgebroid . example 1 ) the case in which the @xmath97 themselves are formed from the generators themselves is well - known . in this case , the association amounts to producing the centre @xmath98 of the _ universal enveloping algebra _ @xmath99 of @xmath33 . this is named thus due to its encapsulation of features universal to all representations of the @xmath33 in question . in this case , the resulting @xmath97 are known as _ casimirs _ these play a prominent role in representation theory , with @xmath100 s total angular momentum operator @xmath101 being the best - known such . in this example of association procedure , each given lie algebra leads to a corresponding algebra of casimirs .          . suppose @xmath17 commutes solely with a set of @xmath3 which is not closed : it does not include some of the @xmath102|}$ ] . however , the jacobi identity with one @xmath103 and two @xmath104 as entries and making two uses of ( [ basic - obs ] ) gives                    a _ poset _ ( _ partially ordered set _ ) is a set @xmath109 alongside an ordering relation @xmath110 which is reflexive , antisymmetric and transitive . this is less stringent than the more familiar total ordering , for which every two elements have to be related ; this permits a linear ranking representation , by which total orderings are also known as _ posets elements need not all be related , by which posets in general contain multiple branching chains . this is representable using a _ hasse diagram _ ; all the figures in the current paper are based on hasse diagrams . an element 1 of @xmath111 is a _ unit _ if @xmath112 , @xmath113 , and an element 0 of @xmath111 is a _ null element _ if @xmath112 , @xmath114 . a lattice possessing these is called a _ bounded lattice _ ; this applies to all examples in the current article . subset inclusion @xmath115 is one common realization of @xmath110 . this carries over to the case of sets with further structures , in which case one is dealing with subspace inclusion . in this setting , intersection of two subspaces plays the role of meet and smallest subspace containing a pair of spaces that of their join . e.g. for subgroups of a group @xmath14 , the intersection of two subgroups is also a subgroup , and the join of two subgroups is defined as the subgroup generated by their union . 0 = i d : the trivial group , and 1 = @xmath14 itself . constraint subalgebraic structures of an algebraic structure @xmath116 follow suit in this regard , with 0 = i d : the trivial algebra , and 1 = @xmath116 itself . on the other hand , for observables subalgebraic structures of an observables algebraic structure @xmath117 associated a with constraint algebraic structure @xmath116 , each constraint present in the latter induces an extra relation in the former . hence the association map @xmath118 is an order - reversing lattice morphism @xmath119 . this can be recast as order - preserving by use of @xmath120 in place of @xmath115 . @xmath117 still possesses a 0 and a 1 , with , by the order - reversing , @xmath121 the space @xmath122 of unconstrained observables @xmath6 and @xmath123 the space @xmath124 maximally constrained i.e . observables @xmath5 . hence the lattice of observables is also a bounded lattice . note that @xmath118 maps all constraint subalgebraic structures to observables algebraic substructures , since one can always pose the problem of what functions commute with a given constraint subalgebraic structure . adding constraints can never increase the freedom in the observables algebraic structure , but on occasion it may have no effect if the constraint being added imposes no further restriction on the allowed functional form of the observables . e.g. an angle quantity is already a ratio and a combination of dot products . i.e. @xmath118 is capable of being many to one , as occurs in the example of conformal relational mechanics . in such a case , @xmath118 is a coarse - graining of the ordering information in moving between the two lattices ; also note that in all the other examples considered in this article , @xmath118 is 1 : 1 and fully preserving of the ordering information . see @xcite for a bit more about lattices with basic and theoretical physics examples , and @xcite for somewhat more theory . with many conceptually and technically interesting properties of lattices having been worked out , it will be very interesting to see which of these apply to lattices of a - observables . kucha ,  time and interpretations of quantum gravity " , in _ proceedings of the 4th canadian conference on general relativity and relativistic astrophysics _ ed . g. kunstatter , d. vincent and j. williams ( world scientific , singapore , 1992 ) , reprinted as int d20 * 3 ( 2011 ) . canonical quantum gravity and the problem of time " , in _ integrable systems , quantum groups and quantum field theories _ ed . ibort and m.a . rodrguez ( kluwer , dordrecht 1993 ) , gr - qc/9210011 . kucha ,  canonical quantum gravity " , in _ general relativity and gravitation 1992 _ , ed gleiser , c.n . kozamah and o.m . moreschi m ( institute of physics publishing , bristol 1993 ) , gr - qc/9304012 . c. rovelli , `` is there incompatibility between the ways time is treated in general relativity and in standard quantum mechanics ? '' , p. 126 in _ conceptual problems of quantum gravity _ ed . a. ashtekar and j. stachel ( birkhuser , boston , 1991 ) . e. anderson ,  the problem of time and quantum cosmology in the relational particle mechanics arena " , arxiv:1111.1472 . e. anderson ,  problem of time in quantum gravity " , annalen der physik , * 524 * 757 ( 2012 ) , arxiv:1206.2403 . e. anderson ,  where to apply relationalism " , write - up of invited seminar at the conferences at fredericton 2014 , arxiv:1411.4316 . r. gilmore , _ lie groups , lie algebras , and some of their applications _ ( dover , new york 2006 ) . v. gorbounov , f. malikov and v. schechtman ,  gerbes of chiral differential operators . iii " , in _ the orbit method in geometry and physics : in honor of a.a . , ed . c. duval , l. guieu , v. ovsienko ( a.m.s . , providence , rhode island 2001 ) .
the observables notion is shown to apply only to a limited range of theories . 
 relational mechanics , slightly inhomogeneous cosmology and supergravity are used as examples that require further notions of observables . a suitably general notion of a - observables 
 is then given to cover all of these cases . ` 
 a ' here stands for ` algebraic substructure ' ; a - observables can be defined by association with each closed algebraic substructure of a theory s constraints . 
 both constrained algebraic structures and associated notions of a - observables form bounded lattices .    * edward anderson *    _ damtp , centre for mathematical sciences , wilberforce road , cambridge cb3 owa . _