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The commutative residuated lattices were first introduced by M. Ward and R.P. Dilworth as generalization of ideal lattices of rings. Complete studies on residuated lattices were developed by H. Ono, T. Kowalski, P. Jipsen and C. Tsinakis. Also, the concept of lattice implication algebra is due to Y. Xu. And Luitzen Brouwer founded the mathematical philosophy of intuitionism, which believed that a statement could only be demonstrated by direct proof. Arend Heyting, a student of Brouwer’s, formalized this thinking into his namesake algebras. In this paper, we investigate the relationship between implicative algebras, Heyting algebras and residuated lattices. In fact, we show that implicative algebras and Heyting algebras can be described as residuated lattices.

The Homrit Waggat granite is a composite granite pluton intruded in metamorphosed volcano-sedimentary association and the metagabbro-diorite complex to the east and north and tonalite-granodiorite suite to the south and northeast. Mineralogically and geochemically the granite phases change from subsolvus peraluminous granodiorite to hypersolvus metaluminous and highly evolved alkali feldspar granite, passing through biotite and mylonitized biotite granites. Late to post-magmatic processes are represented by marginal stockscheider amazonite pegmatite, marginal amazonite albite as well as greisen zone up to 30 m2. Increasing of SiO2, alkalis, Rb, F, Nb, Ta, Sn, Ga, HREEs and Y, and decreasing of Fe, Al, Mg, Ca, Mn, Ti, Sr, Ba, Zr, and LREEs from the biotite granodiorite to hypersolvus alkali feldspar granite reflects magmatic fractionation processes of Homrit Waggat granite phases. LREEs fractionation patterns as well as Eu anomalies decrease from granodiorite to the alkali feldspar granite and the latter displays flat patterns. In the hypersolvus alkali feldspar granite, Ga/Al ratio is typically of A-type granite, but not their Zr, Y, or Ce enrichments. In addition, the hypersolvus granite is characterized by low-P2O5 and the LREEs>>HREEs depletion which reflects the initial undersaturation of accessory mineral assemblage that resulted from high concentration of volatiles and/or alkali complexes. The behaviour of REEs and Zr in the mentioned phases is consistent with F- content as well as accessory minerals in the studied granites. Trace elements pattern in the spider diagram show significant depletion in Sr, Ba, P and Ti, and enrichment in Rb, Th and U. The Sr, Ba, P and Ti depletion could be related to fractionation of plagioclase, apatite and ilmenite. Zircon saturation temperature (Tzr) calculated from bulk rock composition for Homrit Waggat granites range between 809°C and 765°C. These values are consistent with low temperature granite which crystallized from a source melt saturated with zirconium concentrations via partial melting of I- type granite magma may be granodioritic in composition. The highly evolved alkali feldspar granite was formed from the initial granodioritic I-type melt via fractional crystallization. F-rich melt and F-complexing played an important role in the evolution and chemical characterization of the highly evolved hypersolvus alkali feldspar granite. Four stages of mineralization were detected in the Homrit Waggat granite. These stages are magmatic, pegmatitic, metasomatic and veins. Columbite, cassiterite, fluorite as well as undifferentiated rare earth minerals are detected.

The Homrit Waggat granite is a composite granite pluton intruded in metamorphosed volcano-sedimentary association and the metagabbro-diorite complex to the east and north and tonalite-granodiorite suite to the south and northeast. Mineralogically and geochemically the granite phases change from subsolvus peraluminous granodiorite to hypersolvus metaluminous and highly evolved alkali feldspar granite, passing through biotite and mylonitized biotite granites. Late to post-magmatic processes are represented by marginal stockscheider amazonite pegmatite, marginal amazonite albite as well as greisen zone up to 30 m2. Increasing of SiO2, alkalis, Rb, F, Nb, Ta, Sn, Ga, HREEs and Y, and decreasing of Fe, Al, Mg, Ca, Mn, Ti, Sr, Ba, Zr, and LREEs from the biotite granodiorite to hypersolvus alkali feldspar granite reflects magmatic fractionation processes of Homrit Waggat granite phases. LREEs fractionation patterns as well as Eu anomalies decrease from granodiorite to the alkali feldspar granite and the latter displays flat patterns. In the hypersolvus alkali feldspar granite, Ga/Al ratio is typically of A-type granite, but not their Zr, Y, or Ce enrichments. In addition, the hypersolvus granite is characterized by low-P2O5 and the LREEs>>HREEs depletion which reflects the initial undersaturation of accessory mineral assemblage that resulted from high concentration of volatiles and/or alkali complexes. The behaviour of REEs and Zr in the mentioned phases is consistent with F- content as well as accessory minerals in the studied granites. Trace elements pattern in the spider diagram show significant depletion in Sr, Ba, P and Ti, and enrichment in Rb, Th and U. The Sr, Ba, P and Ti depletion could be related to fractionation of plagioclase, apatite and ilmenite. Zircon saturation temperature (Tzr) calculated from bulk rock composition for Homrit Waggat granites range between 809°C and 765°C. These values are consistent with low temperature granite which crystallized from a source melt saturated with zirconium concentrations via partial melting of I- type granite magma may be granodioritic in composition. The highly evolved alkali feldspar granite was formed from the initial granodioritic I-type melt via fractional crystallization. F-rich melt and F-complexing played an important role in the evolution and chemical characterization of the highly evolved hypersolvus alkali feldspar granite. Four stages of mineralization were detected in the Homrit Waggat granite. These stages are magmatic, pegmatitic, metasomatic and veins. Columbite, cassiterite, fluorite as well as undifferentiated rare earth minerals are detected.

The Homrit Waggat granite is a composite granite pluton intruded in metamorphosed volcano-sedimentary association and the metagabbro-diorite complex to the east and north and tonalite-granodiorite suite to the south and northeast. Mineralogically and geochemically the granite phases change from subsolvus peraluminous granodiorite to hypersolvus metaluminous and highly evolved alkali feldspar granite, passing through biotite and mylonitized biotite granites. Late to post-magmatic processes are represented by marginal stockscheider amazonite pegmatite, marginal amazonite albite as well as greisen zone up to 30 m2. Increasing of SiO2, alkalis, Rb, F, Nb, Ta, Sn, Ga, HREEs and Y, and decreasing of Fe, Al, Mg, Ca, Mn, Ti, Sr, Ba, Zr, and LREEs from the biotite granodiorite to hypersolvus alkali feldspar granite reflects magmatic fractionation processes of Homrit Waggat granite phases. LREEs fractionation patterns as well as Eu anomalies decrease from granodiorite to the alkali feldspar granite and the latter displays flat patterns. In the hypersolvus alkali feldspar granite, Ga/Al ratio is typically of A-type granite, but not their Zr, Y, or Ce enrichments. In addition, the hypersolvus granite is characterized by low-P2O5 and the LREEs>>HREEs depletion which reflects the initial undersaturation of accessory mineral assemblage that resulted from high concentration of volatiles and/or alkali complexes. The behaviour of REEs and Zr in the mentioned phases is consistent with F- content as well as accessory minerals in the studied granites. Trace elements pattern in the spider diagram show significant depletion in Sr, Ba, P and Ti, and enrichment in Rb, Th and U. The Sr, Ba, P and Ti depletion could be related to fractionation of plagioclase, apatite and ilmenite. Zircon saturation temperature (Tzr) calculated from bulk rock composition for Homrit Waggat granites range between 809°C and 765°C. These values are consistent with low temperature granite which crystallized from a source melt saturated with zirconium concentrations via partial melting of I- type granite magma may be granodioritic in composition. The highly evolved alkali feldspar granite was formed from the initial granodioritic I-type melt via fractional crystallization. F-rich melt and F-complexing played an important role in the evolution and chemical characterization of the highly evolved hypersolvus alkali feldspar granite. Four stages of mineralization were detected in the Homrit Waggat granite. These stages are magmatic, pegmatitic, metasomatic and veins. Columbite, cassiterite, fluorite as well as undifferentiated rare earth minerals are detected.

The Homrit Waggat granite is a composite granite pluton intruded in metamorphosed volcano-sedimentary association and the metagabbro-diorite complex to the east and north and tonalite-granodiorite suite to the south and northeast. Mineralogically and geochemically the granite phases change from subsolvus peraluminous granodiorite to hypersolvus metaluminous and highly evolved alkali feldspar granite, passing through biotite and mylonitized biotite granites. Late to post-magmatic processes are represented by marginal stockscheider amazonite pegmatite, marginal amazonite albite as well as greisen zone up to 30 m2. Increasing of SiO2, alkalis, Rb, F, Nb, Ta, Sn, Ga, HREEs and Y, and decreasing of Fe, Al, Mg, Ca, Mn, Ti, Sr, Ba, Zr, and LREEs from the biotite granodiorite to hypersolvus alkali feldspar granite reflects magmatic fractionation processes of Homrit Waggat granite phases. LREEs fractionation patterns as well as Eu anomalies decrease from granodiorite to the alkali feldspar granite and the latter displays flat patterns. In the hypersolvus alkali feldspar granite, Ga/Al ratio is typically of A-type granite, but not their Zr, Y, or Ce enrichments. In addition, the hypersolvus granite is characterized by low-P2O5 and the LREEs>>HREEs depletion which reflects the initial undersaturation of accessory mineral assemblage that resulted from high concentration of volatiles and/or alkali complexes. The behaviour of REEs and Zr in the mentioned phases is consistent with F- content as well as accessory minerals in the studied granites. Trace elements pattern in the spider diagram show significant depletion in Sr, Ba, P and Ti, and enrichment in Rb, Th and U. The Sr, Ba, P and Ti depletion could be related to fractionation of plagioclase, apatite and ilmenite. Zircon saturation temperature (Tzr) calculated from bulk rock composition for Homrit Waggat granites range between 809°C and 765°C. These values are consistent with low temperature granite which crystallized from a source melt saturated with zirconium concentrations via partial melting of I- type granite magma may be granodioritic in composition. The highly evolved alkali feldspar granite was formed from the initial granodioritic I-type melt via fractional crystallization. F-rich melt and F-complexing played an important role in the evolution and chemical characterization of the highly evolved hypersolvus alkali feldspar granite. Four stages of mineralization were detected in the Homrit Waggat granite. These stages are magmatic, pegmatitic, metasomatic and veins. Columbite, cassiterite, fluorite as well as undifferentiated rare earth minerals are detected.

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