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varieties
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=== Varieties of universal algebras === A \emph{variety} is a class of structures of the same signature th... 4, 1935)] a class $\mathcal{K}$ of algebras is a variety iff it is closed under the operators $H$, $S$, $P... $\mathcal{K}\}$. Equivalently, $\mathcal K$ is a variety iff $\mathcal K=HSP\mathcal K$. In particular, g... bras, $V\mathcal K=HSP\mathcal K$ is the smallest variety that contains $\mathcal K$, and is called the \em
monounary_algebras
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rties==== ^[[Classtype]] |variety | ^[[Equational theory]] |decidabl... [[Idempotent monounary algebras]] subvariety The variety of monounary algebras has countably many subvarie... ttice of divisibility of the natural numbers. The variety $\text{Mod}(x=y)$ of trivial subvarieties is the unique element below the variety $\text{Mod}(f(x)=x)$ (which is term-equivalent to
boolean_semilattices
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g,\cdot\rangle$ such that $\mathbf{A}$ is in the variety generated by complex algebras of semilattices Le... results==== ====Properties==== ^[[Classtype]] |variety | ^[[Finitely axiomatizable]] |open | ^[[Equatio... [[Some members of BSlat]] ====Subclasses==== [[Variety generated by complex algebras of linear semilatti
semilattices
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: This definition shows that semilattices form a variety. ==Morphisms== Let $\mathbf{S}$ and $\mathbf{T}$... shows that semilattices form a partially-ordered variety. ====Definition==== A \emph{join-semilattice} is ... results==== ====Properties==== ^[[Classtype]] |variety | ^[[Equational theory]] |decidable in polynomia
normal_valued_lattice-ordered_groups
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e$ ====Examples==== ====Basic results==== The variety of normal valued $\ell$-groups is the largest pro... perties==== ^[[Classtype]] |variety | ^[[Equational theory]] | | ^[[Qua... 976> W. Charles Holland, \emph{The largest proper variety of lattice-ordered groups}, Proceedings of the AM
right_hoops
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k: This definition shows that right hoops form a variety. Right hoops are partially ordered by the relati... results==== ====Properties==== ^[[Classtype]] |variety | ^[[Equational theory]] | | ^[[Quasiequational
kleene_logic_algebras
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sults==== The algebra in Example 1 generates the variety of Kleene logic algebras ====Properties==== ^[[Classtype]] |Variety | ^[[Equational theory]] |Decidable | ^[[Quasieq
modal_algebras
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results==== ====Properties==== ^[[Classtype]] |variety | ^[[Equational theory]] |decidable | ^[[Quasieq... tionally def. pr. cong.]] |no | ^[[Discriminator variety]] |no | ^[[Amalgamation property]] |yes | ^[[St
equationally_def._pr._cong
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ly definable principal congruences===== A (quasi)variety $\mathcal{K}$ of algebraic structures has \emph{e... roperties that imply EDP(R)C === [[Discriminator variety]] === Properties implied by EDP(R)C === Relativ
sqrt-quasi-mv-algebras
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ac12,\frac12\rangle$. ====Basic results==== The variety of $\sqrt{'}$qMV-algebras is generated by the sta... -algebra]]. ====Properties==== ^[[Classtype]] |variety | ^[[Equational theory]] |decidable | ^[[Quasieq
abelian_groups
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, with addition, unary subtraction, and zero. The variety of abelian groups is generated by this algebra. ... perties==== ^[[Classtype]] |variety | ^[[Equational theory]] |decidable
congruence_n-permutable
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aracterized by a Mal'cev condition. For $n=2$, a variety is congruence permutable iff there exists a term ... the identities $p(x,z,z)=x=p(z,z,x)$ hold in the variety. === Properties that imply congruence $n$-permut
tense_algebras
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results==== ====Properties==== ^[[Classtype]] |variety | ^[[Equational theory]] |decidable | ^[[Quasieq... tionally def. pr. cong.]] |no | ^[[Discriminator variety]] |no | ^[[Amalgamation property]] |yes | ^[[St
function_rings
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y)$. ====Examples==== ====Basic results==== The variety of $f$-rings is generated by the class of linearl... ory, 1967]). ====Properties==== ^[[Classtype]] |variety | ^[[Equational theory]] | | ^[[Quasiequational
sequential_algebras
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results==== ====Properties==== ^[[Classtype]] |variety | ^[[Equational theory]] |undecidable | ^[[Quasi... ionally def. pr. cong.]] |yes | ^[[Discriminator variety]] |no | ^[[Amalgamation property]] |no | ^[[Str
closure_algebras
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de_morgan_algebras
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lattices
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nonassociative_relation_algebras
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quasi-mv-algebra
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hoops
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relation_algebras
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hilbert_algebras
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boolean_groups
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distributive_lattice_ordered_semigroups
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boolean_semigroups
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abelian_lattice-ordered_groups
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commutative_lattice-ordered_semigroups
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idempotent_semirings_with_zero
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idempotent_semirings_with_identity_and_zero
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boolean_algebras
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semirings_with_identity_and_zero
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pseudo_mv-algebras
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residuated_lattices
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commutative_bck-algebras
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directed_graphs
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normal_bands
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tarski_algebras
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multiplicative_additive_linear_logic_algebras
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skew_lattices
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properties
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rings
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lie_algebras
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basic_logic_algebras
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multiplicative_semilattices
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bck-meet-semilattices
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action_algebras
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generalized_boolean_algebras
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dunn_monoids
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de_morgan_monoids
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boolean_monoids
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lattice-ordered_rings
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monoidal_t-norm_logic_algebras
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commutative_lattice-ordered_monoids
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groups
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fl-algebras
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implicative_lattices
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wajsberg_hoops
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integral_relation_algebras
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commutative_inverse_semigroups
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pseudocomplemented_distributive_lattices
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bck-algebras
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distributive_residuated_lattices
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ockham_algebras
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near-rings
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orthomodular_lattices
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semigroups_with_zero
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distributive_double_p-algebras
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semigroups
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clifford_semigroups
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near-rings_with_identity
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lukasiewicz_algebras_of_order_n
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semirings_with_zero
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vector_spaces
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commutative_rings
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almost_distributive_lattices
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bck-lattices
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commutative_residuated_lattices
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bands
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bilinear_algebras
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commutative_groupoids
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quasigroups
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commutative_rings_with_identity
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rings_with_identity
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cancellative_residuated_lattices
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representable_lattice-ordered_groups
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ortholattices
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sets
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reflexive_relations
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bilattices
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binars
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congruence_distributive
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commutative_binars
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congruence_extension_property
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m-zeroid
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quandles
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2-element_boolean_algebra
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classtype
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right_quasigroups
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categories
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brouwerian_semilattices
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modular_lattices
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directoids
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integral_residuated_lattices
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neardistributive_lattices
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vector_spaces_over_a_field
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distributive_lattices_with_operators
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commutative_monoids
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semirings_with_identity
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bounded_distributive_lattices
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double_stone_algebras
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commutative_semigroups
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residuated_partially_ordered_monoids
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medial_groupoids
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unary_algebras
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flec-algebras
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monoids
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generalized_mv-algebras
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loops
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goedel_algebras
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lattice-ordered_semigroups
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bck-join-semilattices
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moufang_loops
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bounded_lattices
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lattice-ordered_monoids
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boolean_algebras_with_operators
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m-sets
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medial_quasigroups
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shells
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boolean_modules_over_a_relation_algebra
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semilattices_with_zero
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modules_over_a_ring
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bounded_residuated_lattices
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rectangular_bands
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idempotent_semirings
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distributive_lattices
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mv-algebras
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cylindric_algebras
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commutative_residuated_lattice-ordered_semigroups
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heyting_algebras
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distributive_dual_p-algebras
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definable_principal_congruences
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idempotent_semirings_with_identity
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multiplicative_lattices
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inverse_semigroups
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complemented_modular_lattices
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fle-algebras
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action_lattices
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semirings
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lattice-ordered_groups
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generalized_bl-algebras
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distributive_p-algebras
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involutive_lattices
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monadic_algebras
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semilattices_with_identity
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order_algebras
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