The Enantioselectivity and the Stereochemical Course of Copper‐Catalyzed Intramolecular CH Insertions of Phenyliodonium Ylides

The Cu‐catalyzed intramolecular CH insertion of phenyliodonium ylide 1b was investigated at 0° in the presence of several chiral ligands. Enantioselectivities varied in the range 38–72%, and were higher than those resulting from reaction of the diazo compound 1c at 65°. The intramolecular insertion of the enantiomerically pure methyl diazoacetate (R)‐ 20 and of the corresponding phenyliodonium ylide (R)‐ 21 proceeded to (R)‐ 23 with retention of configuration with [Cu(hfa)₂] (hfa=hexafluoroacetylacetone=1,1,1,5,5,5‐hexafluoropentane‐2,4‐dione) and [Rh₂(OAc)₄]. These results are consistent with a carbenoid mechanism for the Cu‐catalyzed insertion with phenyliodonium ylides. However, the insertion of the perfluorosulfonated phenyliodonium ylide (R)‐ 29 afforded with [Cu(hfa)₂] as well as with [Rh₂(OAc)₄] the cyclopentanone derivative 30 as a cis/trans mixture with only 56–67% enantiomeric excess.     

Enantioselectivity and cis/trans-Selectivity in Dirhodium(II)-Catalyzed addition of diazoacetates to olefins

The Rh¹¹-catalyzed carbenoid addition of diazoacetates to olefins was investigated with [Rh₂{(4S)-phox}₄] ( 1 ;phox = tetrakis[(4S)-tetrahydro-4-phenyloxazol-2-one]), [Rh₂{(2S)-mepy}₄] ( 2 ; mepy = tetrakis[methyl (2S)-tetrahydro-5-oxopyrrole-2-carboxylate]), and [Rh₂(OAc)₄] ( 3 ). While catalysis with 2 and 3 afford preferentially trans-cyclopropanecarboxylates, the cis-isomers are the major products with 1 . In general, the enantioselectivities achieved with 1 and 2 are comparable. Additions catalyzed by 1 are strongly sensitive to steric effects. Highly substituted olefins afford cyclopropanes in only poor yield. The preferential cis-selectivity observed in reactions catalyzed by 1 is attributed to dominant interactions between the ligand of the catalyst and the substituents of both olefin and diazoacetate, which overrule the steric interactions between olefin and diazoacetate in the transition state for carbene transfer.     

cis-Disubstituted Cyclopropanes via Asymmetric Catalytic Cyclopropenation: Synthesis of Cyclopropyl-dehydroamino Acids and of Dictyopterene C′

The cyclopropenation of diethoxypropyne ( 1 ) with methyl diazoacetate in the presence of [Rh₂{(2S)-mepy}₄] (mepy=methyl 5-oxopyrrolidine-2-carboxylate) proceeds with >95% ee. The resulting cyclopropenecarboxylate 2 underwent stereoselective hydrogenation to the cis-cyclopropane 3 . Hydrolysis of the acetal function of 3 liberated the formyl cyclopropenecarboxylate 4 , which was transformed by Wittig reaction with the phosphonate 5 to afford dehydroamino acid 6 as a mixture of (Z)- and (E)-isomers in various proportions. The (Z)-isomer 6a was hydrolyzed, and the structure and the absolute configuration of the (Z)-dicarboxylic acid 7a were established by X-ray crystallography. The cis-divinylcyclopropane 11 (ee>95%), in turn, was synthesized from 4 via Wittig reaction to afford 8 , which was transformed to the aldehyde 10 and subjected to a second Wittig reaction. Thermolysis of 11 afforded (+)-dictyopterene C′ ( 12 ) in quantitative yield.     

Carbenoid Reactions in Rhodium(II)-Catalyzed Decomposition of Iodonium Ylides

The intermediacy of metallocarbenes in decomposition reactions of iodonium ylides with [Rh₂(OAc)₄] was established by comparison with reactions of the corresponding diazo compounds. The sensitivity of the Rh^II-catalyzed intermolecular cyclopropane formation from substituted styrenes and bis(methoxycarbonyl)(phenyliodono)methanide ( 1a ) or dimethyl diazomalonate ( 1b ) is identical. The Hammett plot (with σ⁺) has a slope of ?0.47. Iodonium ylides and diazo compounds afford the same products in [Rh₂(OAc)₄]-catalyzed cyclopropane formations, cycloadditions, and intramolecular CH insertions, and exhibit the identical selectivity in intramolecular competitions for cyclopropane formation and insertion. The intramolecular CH insertion of the ylide 20c , when carried out in the presence of a chiral catalyst ([Rh₂{(?)-(S)-ptpa}₄]), results in formation of 21a having an ee of 67%, identical to the ee obtained with the diazo compound 20b .     

Borane-Catalyzed Tandem Cyclization/Hydrosilylation Towards Enantio- and Diastereoselective Construction of trans-2,3-Disubstituted-1,2,3,4-Tetrahydroquinoxalines

Recent years have witnessed marked progress in the efficient synthesis of various enantioenriched 1,2,3,4-tetrahydroquinoxalines. However, enantio- and diastereoselective access to trans-2,3-disubstituted 1,2,3,4-tetrahydroquinoxalines remains much less explored. Herein we report that a frustrated Lewis pair-based catalyst generated via in situ hydroboration of 2-vinylnaphthalene with HB(C₆F₅)₂ allows for the one-pot tandem cyclization/hydrosilylation of 1,2-diaminobenzenes and 1,2-diketones with commercially available PhSiH₃ to exclusively afford trans-2,3-disubstituted 1,2,3,4-tetrahydroquinoxalines in high yields with excellent diastereoselectivities (>20 : 1 dr). Furthermore, this reaction can be rendered asymmetric by using an enantioenriched borane-based catalyst derived from HB(C₆F₅)₂ and a binaphthyl-based chiral diene to give rise to enantioenriched trans-2,3-disubstituted 1,2,3,4-tetrahydroquinoxalines in high yields with almost complete diastereo- and enantiocontrol (>20 : 1 dr, up to >99 % ee). A wide substrate scope, good tolerance of diverse functionality and up to 20-gram scale production are demonstrated. The enantio- and diastereocontrol are achieved by the judicious choice of borane catalyst and hydrosilane. The catalytic pathway and the origin of the excellent stereoselectivity are elucidated by mechanistic experiments and DFT calculations.     

Combined Experimental and Computational Investigations of Rhodium‐Catalysed CH Functionalisation of Pyrazoles with Alkenes

Detailed experimental and computational studies have been carried out on the oxidative coupling of the alkenes C₂H₃Y (Y=CO₂Me ( a ), Ph ( b ), C(O)Me ( c )) with 3‐aryl‐5‐R‐pyrazoles (R=Me ( 1 a ), Ph ( 1 b ), CF₃ ( 1 c )) using a [Rh(MeCN)₃Cp*][PF₆]₂/Cu(OAc)₂ ? H₂O catalyst system. In the reaction of methyl acrylate with 1 a , up to five products ( 2 aa – 6 aa ) were formed, including the trans monovinyl product, either complexed within a novel Cu^I dimer ( 2 aa ) or as the free species ( 3 aa ), and a divinyl species ( 6 aa ); both 3 aa and 6 aa underwent cyclisation by an aza‐Michael reaction to give fused heterocycles 4 aa and 5 aa , respectively. With styrene, only trans mono‐ and divinylation products were observed, whereas with methyl vinyl ketone, a stronger Michael acceptor, only cyclised oxidative coupling products were formed. Density functional theory calculations were performed to characterise the different migratory insertion and β‐H transfer steps implicated in the reactions of 1 a with methyl acrylate and styrene. The calculations showed a clear kinetic preference for 2,1‐insertion and the formation of trans vinyl products, consistent with the experimental results.     

Elevated Catalytic Activity of Ruthenium(II)–Porphyrin‐Catalyzed Carbene/Nitrene Transfer and Insertion Reactions with N‐Heterocyclic Carbene Ligands

Bis(NHC)ruthenium(II)–porphyrin complexes were designed, synthesized, and characterized. Owing to the strong donor strength of axial NHC ligands in stabilizing the trans M?CRR′/M?NR moiety, these complexes showed unprecedently high catalytic activity towards alkene cyclopropanation, carbene C? H, N? H, S? H, and O? H insertion, alkene aziridination, and nitrene C? H insertion with turnover frequencies up to 1950 min^?1. The use of chiral [Ru(D₄‐Por)(BIMe)₂] ( 1 g ) as a catalyst led to highly enantioselective carbene/nitrene transfer and insertion reactions with up to 98 % ee. Carbene modification of the N terminus of peptides at 37 °C was possible. DFT calculations revealed that the trans axial NHC ligand facilitates the decomposition of diazo compounds by stabilizing the metal–carbene reaction intermediate.     

A 1,1‐Carboboration Route to Bora‐Nazarov Systems

Hydroboration of the conjugated enynes 1 a and 1 b with Piers’ borane [HB(C₆F₅)₂] gave the respective dienylboranes trans‐ 2 c and trans‐ 2 d . Their photolysis resulted in the formation of the dihydroborole products 3 c and 3 d . Both were converted to their pyridine adducts 5 c and 5 d , respectively. Compounds 3 c and 5 c,d were characterized by X‐ray diffraction. The reaction of the bis(enynyl)boranes 6 a and 6 b with B(C₆F₅)₃ resulted in the formation of the dihydroboroles 7 a and 7 b , respectively. This reaction is thought to proceed by 1,1‐carboboration of one of the enynyl substituents at boron to generate the dienylborane intermediates 8 a / 8 b , followed by thermally induced bora‐Nazarov ring‐closure and subsequent stabilizing 1,2‐pentafluorophenyl group migration from boron to carbon. Compound 7 a was characterized by X‐ray diffraction and solid‐state ¹¹B NMR spectroscopy.     

Syntheses and Structures of Ruthenium(II) N,S‐Heterocyclic Carbene Diphosphine Complexes and their Catalytic Activity towards Transfer Hydrogenation

Phosphine exchange of [Ru^IIBr(MeCOO)(PPh₃)₂(3‐RBzTh)] (3‐RBzTh=3‐benzylbenzothiazol‐2‐ylidene) with a series of diphosphines (bis(diphenylphosphino)methane (dppm), 1,2‐bis(diphenylphosphino)ethylene (dppv), 1,1′‐bis(diphenylphosphino)ferrocene (dppf), 1,4‐bis(diphenylphosphino)butane (dppb), and 1,3‐(diphenylphosphino)propane (dppp)) gave mononuclear and neutral octahedral complexes [RuBr(MeCOO)(η²‐P₂)(3‐RBzTh)] (P₂=dppm ( 2 ), dppv ( 3 ), dppf ( 4 ), dppb ( 5 ), or dppp ( 6 )), the coordination spheres of which contained four different ligands, namely, a chelating diphosphine, carboxylate, N,S‐heterocyclic carbene (NSHC), and a bromide. Two geometric isomers of 6 ( 6a and 6 b ) have been isolated. The structures of these products, which have been elucidated by single‐crystal X‐ray crystallography, show two structural types, I and II, depending on the relative dispositions of the ligands. Type I structures contain a carbenic carbon atom trans to the oxygen atom, whereas two phosphorus atoms are trans to bromine and oxygen atoms. The type II system comprises a carbene carbon atom trans to one of the phosphorus atoms, whereas the other phosphorus is trans to the oxygen atom, with the bromine trans to the remaining oxygen atom. Complexes 2 , 3 , 4 , and 6a belong to type I, whereas 5 and 6 b are of type II. The kinetic product 6 b eventually converts into 6a upon standing. These complexes are active towards catalytic reduction of para‐methyl acetophenone by 2‐propanol at 82 °C under 1 % catalyst load giving the corresponding alcohols. The dppm complex 2 shows the good yields (91–97 %) towards selected ketones.     

Enantiomer differentiation in intramolecular carbon—hydrogen insertion reactions of racemic secondary alkyl diazoacetates catalyzed by chiral dirhodium(ii) carboxamidates

Highly efficient kinetic resolution of racemic secondary alkyl diazoacetates in intramolecular carbon-hydrogen insertion reactions has been achieved using chiral dirhodium(ii) carboxamidates. Products formed from catalytic diazo decomposition of racemic 2-octyl diazoacetate and, separately, its (2R)- and (2S)-enantiomeric forms, as well as bothcis- andtrans-2-methylcyclohexyl diazoacetates, have been systematically evaluated. Enantioselectivities up to 99 %ee have been obtained for -lactone formation. -Lactone production has been observed and, although minor with cyclohexyl diazoacetates, is the major insertion pathway for diazo decomposition of 2-octyl diazoacetate.Dedicated to Academician of the RAS N. S. Zefirov (on his 60th birthday).Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1798–1803, September, 1995.Financial support for this research from the National Institutes of Health (GM 46503) and the National Science Foundation of the United States is gratefully acknowledged. We thank D. A. Pierson for her preparation of 2-methylcyclohexyl diazoacetates and preliminary studies of their diazo decomposition and A. Melekhov from the Higher College of Chemistry for his preparation and catalytic studies of rac-2-octyl diazoacetate.     

Titanium complexes of dialkanolamine ligands as initiators for living ring‐opening polymerization of ε‐caprolactone

The titanium complexes with one ( 1a , 1b , 1c ) and two ( 2a , 2b ) dialkanolamine ligands were used as initiators in the ring‐opening polymerization (ROP) of ε‐caprolactone. Titanocanes 1a and 1b initiated living ROP of ε‐caprolactone affording polymers whose number‐average molecular weights (M_n) increased in direct proportion to monomer conversion (M_n ≤ 30,000 g mol^?1) in agreement with calculated values, and were inversely proportional to initiator concentration, while the molecular weight distribution stayed narrow throughout the polymerization (M_w/M_n ≤ 1.2 up to 80% monomer conversion). ¹H‐NMR and MALDI‐TOF‐MS studies of the obtained poly(ε‐caprolactone)s revealed the presence of an isopropoxy group originated from the initiator at the polymer termini, indicating that the polymerization takes place exclusively at the Ti–OⁱPr bond of the catalyst. The higher molecular weight polymers (M_n ≤ 70,000 g mol^?1) with reasonable MWD (M_w/M_n ≤ 1.6) were synthesized by living ROP of ε‐caprolactone using spirobititanocanes ( 2a , 2b ) and titanocane 1c as initiators. The latter catalysts, according MALDI‐TOF‐MS data, afford poly(ε‐caprolactone)s with almost equal content of α,ω‐dihydroxyl‐ and α‐hydroxyl‐ω(carboxylic acid)‐terminated chains arising due to monomer insertion into “Ti–O” bond of dialkanolamine ligand and from initiation via traces of water, respectively. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 1230–1240, 2010     

To Coordinate or not to Coordinate: The Special Role of Chalcogen Ether Functionalities in the Design of Twofold Functionalized Cyclopentadienyl Ligands [Cp,O,Ch (Ch = S,Se)]

The solvent‐ and catalyst free synthesis of two β‐thio ketones L1a and L1b is reported. L1a , L1b , and a β‐seleno ketone L1c were successfully employed as ligand precursors in the synthesis of a novel series of cationic titanium complexes 4a – 4c via a well‐established reaction sequence: insertion of the carbonyl functional group into the polarized Ti–C_q,exo bond of the monopentafulvene complex Cp*Ti(Cl)(π‐η⁵:σ–η¹‐C₅H₄=CR₂) ( 1 ) (CR₂ = adamantylidene), subsequent methylation, and final activation with B(C₆F₅)₃. The cationic titanium complexes 4a – 4c bear twofold functionalized cyclopentadienyl [Cp,O,Ch (Ch = S, Se)] ligand frameworks built directly in the coordination sphere of the metal, in which the chalcogen ether functionalities do not coordinate to the central metal atoms as demonstrated by NMR experiments. Consequently, Cp,O σ,π chelating ligand systems are formed with free coordination sites at the central titanium atoms and pendant chalcogen ether moieties.     

Synthese von (±)-α-Chamigren. Vorläufige Mitteilung

Synthesis of (±)-α-Chamigrene Cis- and trans-β-ionol (cis and trans- 1 ) underwent acid catalysed dehydration to a mixture of the tetraenes 2–5 in 70–80% yield (Table 1). Irradiation of this mixtures made the 6-(Z), 8-(Z)-isomer 5 accessible (columns 3 and 4 in Table 1). Pyrolysis of the different mixtures at 170° showed, that both isomers, 3 and 5 respectively undergo electrocyclization to dehydrochamigrene ( 6 ). The latter was reduced to α-chamigrene ( 7 ) by hydrogen on Lindlar catalyst.     

Evidence for Internal Ion Pair Formation upon Insertion of Reactive Alkene in the Zirconium–Carbon Bond of the cp2Zr(μ-C4h6)b(c6f5)3 Metallocene-Boron-Betaine Ziegler Catalyst System

Treatment of the (butadiene)ML₂ complexes 1 [ML₂ = Cp₂Zr ( a ), Cp₂Hf ( b ), and (.-C₅H₄CH₃)₂Zr ( c )] with B(C₆F₅)₃ gives the 1:1 addition products (CH₂CHCHCH₂–B(C₆F₅_₃)ML₂ ( 3a – c ). At –40°C the betaine complex 3a inserts one equivalent of methylenecyclopropane to give the regioisomeric insertion products 5a and 6a in a 60:40 ratio. These products exhibit the cyclopropylidene moiety in the α- and β-positions, respectively, relative to zirconium. The corresponding hafnocene complexes 5b and 6b are obtained in a 70:30 ratio starting from 3b . The reaction of 3 ( a – c ) with allene gives a single insertion product ( 7a – c ) in each case where the exo-methylene group is in the α-position to the metal center ([2,1]-insertion). The complexes 5 – 7 are chiral. They all exhibit a pronounced ·-interaction of the internal –C⁴H=C⁵H⁻ double bond of the s̀-ligand chain with the metal center in addition to a metallocene/–C⁶H₂–[B] ion pair interaction. The relative contributions of the cationic metallocene end of the dipolar complexes 5 – 7 are quite dependent on the steric and electronic properties of the respective metallocene units involved. This is revealed by a comparison to typical ¹³C-NMR parameters of the complexes 5 – 7 with a pair of suitable model complexes, namely the ethylene insertion product 4 into the betaine system 3a and its THF adduct 4 .THF.     

Reactions of Aliphatic Diazo Compounds: V. Reaction of Methyl Diazoacetate with Imides of Itaconic Acid

Methyl diazoacetate regioselectively adds to N-substituted imides of itaconic acid to afford 2-pyrazolines, methyl 7-aryl-6,8-dioxo-1,2,7-triazaspiro[4.4]non-2-ene-3-carboxylates that in reaction with halogens (Cl₂, Br₂) yield methyl 5-aryl-1-halo-4,6-dioxo-5-azaspiro[2.4]heptane-1-carboxylates as a mixture of syn- and anti-isomers.     

Zur oxidativen Addition von 1-Halogenalk-1-inen – Synthese und Struktur von Phenylalkinylpalladium-Komplexen

On the Oxidative Addition of 1-Halogenalk-1-ynes – Synthesis and Structure of Phenylalkynylpalladium Complexes [Pd(PPh₃)₄] ( 2 ) reacts with IC≡CPh and ClC≡CPh in the sense of an oxidative addition to give trans-[Pd(C≡CPh)X(PPh₃)₂] (X = I: 3 a , X = Cl: 3 b ). As side products trans-[PdX₂(PPh₃)₂] (X = I: 4 a , X = Cl: 4 b ; < 10%) and PhC≡C–C≡CPh ( 5 ; X = I: ca 30%, X = Cl: < 4%) are formed. 3 a and 3 b were characterized by NMR (¹H, ¹³C, ³¹P) and IR spectroscopies as well as by X-ray single-crystal structure analyses. In the crystals of 3 a and 3 b isolated molecules were found. The Pd–C≡C–Ph unit is linear in 3 a and approximately linear in 3 b [Pd–C≡C 174.2(6)°, C≡C–C 179,0(7)°].     

Electrochemical Reduction of Cyclohex-2-enones

The electrochemical reduction of the cyclohex-2-enones 1a–1e (mercury cathode, CH₃CN, Bu₄NBF₄) was studied by means of cyclic voltammetry, d.c. polarography, coulometry and chemical product analysis. Compounds 1a–1c give a mixture of the hydrodimers 4 and 5 via formation of the radical anion 2 by an irreversible one electron transfer, followed by protonation and dimerization of the allylic radical 3 . The 6-halocyclohex-2-enones 1d and 1e exhibit two distinct reduction waves. The first corresponds to an irreversible two electron transfer with formation of the halide anion and the enolate anion 6 which gives 1b by protonation. The second wave corresponds to a quasi-reversible one electron transfer to 6 to afford the radical dianion 7 (Scheme 2).     

1,6-Additionen an 3-Methyl-5-methyliden-2-(5H)-furanon-Derivate

1,6-Additions to 3-Methyl-5-methylidene-2(5H)-furanone Derivatives The anions of thiophenol, methyl malonate and malononitrile react with 3-methyl-5,6-dihydro-2 (4H)-benzofuranone ( 1c ) by the formation of the corresponding 1,6-addition products cis- 5c (63%), trans- 6 (42%) and trans- 7 (76%), respectively. Likewise, the reaction of 3-methyl-5-methylidene-2 (5H)-furanone ( 1b ) with thiophenol yields the 1,6-addition product 5b (66%), and with the sodium salt of methyl aceto-acetate the 1,6-addition product 8 (11%) and the dispiro-dilactone 9 (39%).     

Synthesis of Novel Pyrimido[4,5‐b]quinolin‐4‐ones with Potential Antitumor Activity

The 5,6,7,8,9,10‐hexahydro‐2‐methylthiopyrimido[4,5‐b]quinolines 4a , 4b , 4c , 4d , 5a , 5b , 5c , 5d and their oxidized forms 6a , 6b , 6c , 6d , 7a , 7b , 7c , 7d were obtained from the reaction of 6‐amino‐2‐(methylthio)pyrimidin‐4(3H)‐one 2 or 6‐amino‐3‐methyl‐2‐(methylthio)pyrimidin‐4(3H)‐one 3 and α,β‐unsaturated ketones 1a , 1b , 1c , 1d using BF₃.OEt₂ as catalyst and p‐chloranil as oxidizing agent. Some of the new compounds were evaluated in the US National Cancer Institute (NCI), where compound 5a presented remarkable activity against 46 cancer cell lines, with the most important GI₅₀ values ranging from 0.72 to 18.4 μM from in vitro assays.     

Gyroscope‐Like Molecules Consisting of PdX2/PtX2 Rotators within Three‐Spoke Dibridgehead Diphosphine Stators: Syntheses,Substitution Reactions,Structures, and Dynamic Properties

Threefold intramolecular ring‐closing metatheses of trans‐[MCl₂(P{(CH₂)_mCH?CH₂}₃)₂] are effected with Grubbs’ catalyst. Following hydrogenation catalyzed by [RhCl(PPh₃)₃], the title complexes trans‐[MCl₂(P((CH₂)_n)₃P )] (n=2m+2; M/n=Pt/14, 4 c ; Pt/16, 4 d ; Pt/18, 4 e ; Pd/14, 5 c ; Pd/18, 5 e ) and sometimes isomers partly derived from intraligand metathesis, trans‐[MCl₂{P(CH₂)_n(CH₂)_n}P (CH₂)_n)] ( 4′c–e , 5′e ), are isolated. These react with LiBr, NaI, and KCN to give the corresponding MBr₂, MI₂, and M(CN)₂ species (58–99 %). ¹³C NMR data show that the MX₂ moieties rapidly rotate within the diphosphine cage on the NMR timescale, even at ?120 °C. The reaction of 4 c and KSCN gives separable Pt(NCS)₂ and Pt(NCS)(SCN) adducts ( 13 c , 28 %; 14 c , 20 %), and those of 4 c , e and Ph₂Zn give PtPh₂ species ( 15 c , 61 %; 15 e , 90 %). ¹³C NMR spectra of 13 c – 15 c show two sets of CH₂ signals (ca. 2:1 intensity ratios), indicating that MX₂ rotation is no longer rapid. Reactions of 4 c or 4′c and excess NaC?CH afford the free diphosphines P{(CH₂)₁₄}₃P (91 %) and (CH₂)₁₄P (CH₂)₁₄P(C H₂)₁₄ (90 %). The latter has been crystallographically characterized as a bis(BH₃) adduct. The crystal structures of eight complexes with P(CH₂)₁₄P linkages (PtCl₂, PtBr₂, PtI₂, Pt(NCS)₂, PtPh₂, PdCl₂, PdBr₂, PdI₂) and 15 e have been determined, and intramolecular distances analyzed with respect to MX₂ rotation. The conformations of the (CH₂)₁₄ moieties and features of the crystal lattices are also discussed.